Properties

Degree $12$
Conductor $6.693\times 10^{19}$
Sign $1$
Motivic weight $0$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 6·5-s + 6-s − 7-s − 6·10-s − 11-s + 6·13-s + 14-s − 6·15-s − 17-s + 21-s + 22-s − 23-s + 21·25-s − 6·26-s + 6·30-s + 6·31-s + 33-s + 34-s − 6·35-s − 6·39-s − 42-s − 43-s + 46-s − 47-s − 21·50-s + ⋯
L(s)  = 1  − 2-s − 3-s + 6·5-s + 6-s − 7-s − 6·10-s − 11-s + 6·13-s + 14-s − 6·15-s − 17-s + 21-s + 22-s − 23-s + 21·25-s − 6·26-s + 6·30-s + 6·31-s + 33-s + 34-s − 6·35-s − 6·39-s − 42-s − 43-s + 46-s − 47-s − 21·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 13^{6} \cdot 31^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 13^{6} \cdot 31^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(5^{6} \cdot 13^{6} \cdot 31^{6}\)
Sign: $1$
Motivic weight: \(0\)
Character: induced by $\chi_{2015} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 5^{6} \cdot 13^{6} \cdot 31^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.594106851\)
\(L(\frac12)\) \(\approx\) \(2.594106851\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( ( 1 - T )^{6} \)
13 \( ( 1 - T )^{6} \)
31 \( ( 1 - T )^{6} \)
good2 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
3 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
7 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
11 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
17 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
19 \( ( 1 - T )^{6}( 1 + T )^{6} \)
23 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
29 \( ( 1 - T )^{6}( 1 + T )^{6} \)
37 \( ( 1 - T )^{6}( 1 + T )^{6} \)
41 \( ( 1 - T )^{6}( 1 + T )^{6} \)
43 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
47 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
53 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
59 \( ( 1 - T )^{6}( 1 + T )^{6} \)
61 \( ( 1 - T )^{6}( 1 + T )^{6} \)
67 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
71 \( ( 1 - T )^{6}( 1 + T )^{6} \)
73 \( ( 1 - T )^{6}( 1 + T )^{6} \)
79 \( ( 1 - T )^{6}( 1 + T )^{6} \)
83 \( ( 1 - T )^{6}( 1 + T )^{6} \)
89 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
97 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.07712297812359709301636419625, −5.01457607852384019771441982497, −4.65286380893871379178749336942, −4.61913681788934639303492886440, −4.54409887523622501690404521631, −4.47311772906868840213179001396, −3.97276575357474029598654748630, −3.91264759869244484481188417881, −3.88149702926570741084698361077, −3.31776820361824830169502380280, −3.24675194472769517197183161774, −3.16577836995136424423978186844, −3.01900977791589968834662152145, −2.91608484996336079661788366987, −2.72707226269643274988174939378, −2.41675961549851593372635937691, −2.26357805456589487590510749603, −2.24675019914236999261926051050, −1.94733314344904275299982961783, −1.66573551457843871328628517291, −1.48863415467420407122357934209, −1.21301305312917594204842932823, −1.19300894834485647613845907005, −1.05645648762942095662793994098, −0.967595316357987748728055239231, 0.967595316357987748728055239231, 1.05645648762942095662793994098, 1.19300894834485647613845907005, 1.21301305312917594204842932823, 1.48863415467420407122357934209, 1.66573551457843871328628517291, 1.94733314344904275299982961783, 2.24675019914236999261926051050, 2.26357805456589487590510749603, 2.41675961549851593372635937691, 2.72707226269643274988174939378, 2.91608484996336079661788366987, 3.01900977791589968834662152145, 3.16577836995136424423978186844, 3.24675194472769517197183161774, 3.31776820361824830169502380280, 3.88149702926570741084698361077, 3.91264759869244484481188417881, 3.97276575357474029598654748630, 4.47311772906868840213179001396, 4.54409887523622501690404521631, 4.61913681788934639303492886440, 4.65286380893871379178749336942, 5.01457607852384019771441982497, 5.07712297812359709301636419625

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.