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Properties

Label 2-756-28.3-c1-0-2
Degree $2$
Conductor $756$
Sign $-0.588 - 0.808i$
Analytic cond. $6.03669$
Root an. cond. $2.45696$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.940 − 1.05i)2-s + (−0.232 + 1.98i)4-s + (−1.00 + 0.579i)5-s + (0.250 + 2.63i)7-s + (2.31 − 1.62i)8-s + (1.55 + 0.515i)10-s + (1.17 + 0.679i)11-s + 3.61i·13-s + (2.54 − 2.74i)14-s + (−3.89 − 0.922i)16-s + (−6.02 − 3.48i)17-s + (−3.04 − 5.27i)19-s + (−0.917 − 2.12i)20-s + (−0.388 − 1.88i)22-s + (2.38 − 1.37i)23-s + ⋯
L(s)  = 1  + (−0.664 − 0.747i)2-s + (−0.116 + 0.993i)4-s + (−0.448 + 0.258i)5-s + (0.0948 + 0.995i)7-s + (0.819 − 0.573i)8-s + (0.491 + 0.162i)10-s + (0.355 + 0.204i)11-s + 1.00i·13-s + (0.680 − 0.732i)14-s + (−0.973 − 0.230i)16-s + (−1.46 − 0.844i)17-s + (−0.698 − 1.20i)19-s + (−0.205 − 0.475i)20-s + (−0.0828 − 0.401i)22-s + (0.498 − 0.287i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.588 - 0.808i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.588 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.588 - 0.808i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.588 - 0.808i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.157842 + 0.310033i\)
\(L(\frac12)\) \(\approx\) \(0.157842 + 0.310033i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.940 + 1.05i)T \)
3 \( 1 \)
7 \( 1 + (-0.250 - 2.63i)T \)
good5 \( 1 + (1.00 - 0.579i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.17 - 0.679i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.61iT - 13T^{2} \)
17 \( 1 + (6.02 + 3.48i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.04 + 5.27i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.38 + 1.37i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.87T + 29T^{2} \)
31 \( 1 + (2.09 - 3.62i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.83 - 3.17i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 5.72iT - 41T^{2} \)
43 \( 1 + 5.96iT - 43T^{2} \)
47 \( 1 + (6.26 + 10.8i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.95 - 8.57i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.862 + 1.49i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.23 - 1.28i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.37 - 3.68i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 15.8iT - 71T^{2} \)
73 \( 1 + (3.53 + 2.04i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.0869 + 0.0502i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.12T + 83T^{2} \)
89 \( 1 + (2.78 - 1.61i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.23iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.89737588688180346475381005373, −9.543909403583895710153064965748, −9.052050301787804363148365849397, −8.447995282342614763946622321488, −7.14788635079031639419293068661, −6.66795093489220301976394377629, −4.98788206513036711736583588697, −4.09167246852746783127880427536, −2.81135450511768924530117842958, −1.89886229265904877761022270369, 0.21446051176627289257877070285, 1.74735074575312210314915160142, 3.78077405281924267702722608696, 4.59117806809806487499133954816, 5.86983250112137896825007046275, 6.59011349414181885029910899985, 7.72510067974560538231189210826, 8.088366186994205208906097400602, 9.064456074960874756250244483175, 9.975096324158024231542517741815

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