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Properties

Label 2-756-21.17-c3-0-13
Degree $2$
Conductor $756$
Sign $-0.283 - 0.958i$
Analytic cond. $44.6054$
Root an. cond. $6.67873$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.94 + 8.56i)5-s + (2.97 + 18.2i)7-s + (58.1 + 33.5i)11-s + 88.2i·13-s + (34.5 − 59.7i)17-s + (101. − 58.6i)19-s + (−129. + 74.5i)23-s + (13.5 − 23.4i)25-s + 31.8i·29-s + (−14.1 − 8.17i)31-s + (−141. + 115. i)35-s + (−97.2 − 168. i)37-s − 238.·41-s + 50.6·43-s + (110. + 191. i)47-s + ⋯
L(s)  = 1  + (0.442 + 0.766i)5-s + (0.160 + 0.987i)7-s + (1.59 + 0.920i)11-s + 1.88i·13-s + (0.492 − 0.853i)17-s + (1.22 − 0.708i)19-s + (−1.17 + 0.676i)23-s + (0.108 − 0.187i)25-s + 0.203i·29-s + (−0.0820 − 0.0473i)31-s + (−0.685 + 0.559i)35-s + (−0.431 − 0.748i)37-s − 0.908·41-s + 0.179·43-s + (0.343 + 0.594i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.283 - 0.958i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.283 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.283 - 0.958i$
Analytic conductor: \(44.6054\)
Root analytic conductor: \(6.67873\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :3/2),\ -0.283 - 0.958i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.502936021\)
\(L(\frac12)\) \(\approx\) \(2.502936021\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-2.97 - 18.2i)T \)
good5 \( 1 + (-4.94 - 8.56i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-58.1 - 33.5i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 88.2iT - 2.19e3T^{2} \)
17 \( 1 + (-34.5 + 59.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-101. + 58.6i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (129. - 74.5i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 31.8iT - 2.43e4T^{2} \)
31 \( 1 + (14.1 + 8.17i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (97.2 + 168. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 238.T + 6.89e4T^{2} \)
43 \( 1 - 50.6T + 7.95e4T^{2} \)
47 \( 1 + (-110. - 191. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-448. - 258. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-153. + 265. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-529. + 305. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-119. + 206. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 264. iT - 3.57e5T^{2} \)
73 \( 1 + (507. + 293. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (265. + 460. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 989.T + 5.71e5T^{2} \)
89 \( 1 + (-230. - 399. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 92.1iT - 9.12e5T^{2} \)
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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

−9.861417989269994131291968282978, −9.443507903653018077895749889526, −8.796457498351461945317678640533, −7.28690188202096846622935743387, −6.80143347774711999606440611737, −5.90085067116722550588337287336, −4.78772887880411653492145281283, −3.72032703860932030865481883690, −2.40485711729262819010326082937, −1.55604638573764711177103783109, 0.75000739698439760299346567467, 1.40805665923959671736102245343, 3.33263207591357232510690415622, 4.01765292420977885270825977418, 5.38617540550767176341705807630, 5.95699830805315976783111644787, 7.11971002539424978068409416587, 8.192603505740747352226248630971, 8.640220228716435909577883594803, 10.00468354980743890460280913062

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