Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.996 - 0.0873i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.60 − 2.77i)2-s + (−2.69 + 1.32i)3-s + (−3.15 + 5.45i)4-s + (−4.78 − 1.45i)5-s + (7.99 + 5.35i)6-s + (6.02 + 3.55i)7-s + 7.38·8-s + (5.49 − 7.12i)9-s + (3.61 + 15.6i)10-s + (10.5 + 6.09i)11-s + (1.26 − 18.8i)12-s + 8.47i·13-s + (0.220 − 22.4i)14-s + (14.8 − 2.39i)15-s + (0.744 + 1.29i)16-s + (−5.29 + 9.17i)17-s + ⋯
L(s)  = 1  + (−0.802 − 1.38i)2-s + (−0.897 + 0.441i)3-s + (−0.787 + 1.36i)4-s + (−0.956 − 0.291i)5-s + (1.33 + 0.893i)6-s + (0.861 + 0.508i)7-s + 0.923·8-s + (0.610 − 0.791i)9-s + (0.361 + 1.56i)10-s + (0.959 + 0.553i)11-s + (0.105 − 1.57i)12-s + 0.651i·13-s + (0.0157 − 1.60i)14-s + (0.987 − 0.159i)15-s + (0.0465 + 0.0806i)16-s + (−0.311 + 0.539i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0873i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.996 - 0.0873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.996 - 0.0873i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (74, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.996 - 0.0873i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.491771 + 0.0215280i\)
\(L(\frac12)\)  \(\approx\)  \(0.491771 + 0.0215280i\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + (2.69 - 1.32i)T \)
5 \( 1 + (4.78 + 1.45i)T \)
7 \( 1 + (-6.02 - 3.55i)T \)
good2 \( 1 + (1.60 + 2.77i)T + (-2 + 3.46i)T^{2} \)
11 \( 1 + (-10.5 - 6.09i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 - 8.47iT - 169T^{2} \)
17 \( 1 + (5.29 - 9.17i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (10.0 + 17.4i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-15.2 - 26.4i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 42.8iT - 841T^{2} \)
31 \( 1 + (6.11 - 10.5i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (28.8 - 16.6i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 6.40iT - 1.68e3T^{2} \)
43 \( 1 + 20.0iT - 1.84e3T^{2} \)
47 \( 1 + (-11.8 - 20.5i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-43.9 + 76.1i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (41.9 + 24.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-10.4 - 18.1i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (19.6 + 11.3i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 2.44iT - 5.04e3T^{2} \)
73 \( 1 + (-76.5 - 44.2i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (3.20 + 5.54i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 103.T + 6.88e3T^{2} \)
89 \( 1 + (54.2 - 31.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 140. iT - 9.40e3T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−12.78289834520279837000126035591, −11.96646611653701851401629201220, −11.40127354415654204473682633114, −10.72691939635638843647716668602, −9.292145382830567885484375789163, −8.663241851827830389616498200564, −6.96941575731355666252195426650, −4.90798824278793577160553636355, −3.76851446483748331733494666849, −1.45100669491318578020145093475, 0.60341931898030753111712855625, 4.42246698905313280908238081137, 5.89875512605958566997437547016, 6.94926687354583238073807707712, 7.77036899523037491869822771435, 8.626710920325621649760875827130, 10.35140767200621292432602772150, 11.28937090545467403661742589743, 12.27389013219697780439830782826, 13.89439168384238883970868494795

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