# Properties

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

# Origins of factors

## 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}$$
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$$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.