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Properties

Label 2-126-7.2-c5-0-10
Degree $2$
Conductor $126$
Sign $-0.283 + 0.958i$
Analytic cond. $20.2083$
Root an. cond. $4.49537$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2 − 3.46i)2-s + (−7.99 − 13.8i)4-s + (−37.7 + 65.3i)5-s + (99.4 − 83.1i)7-s − 63.9·8-s + (150. + 261. i)10-s + (−74.7 − 129. i)11-s + 349.·13-s + (−88.9 − 510. i)14-s + (−128 + 221. i)16-s + (−574. − 995. i)17-s + (1.39e3 − 2.42e3i)19-s + 1.20e3·20-s − 597.·22-s + (906. − 1.57e3i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.675 + 1.16i)5-s + (0.767 − 0.641i)7-s − 0.353·8-s + (0.477 + 0.826i)10-s + (−0.186 − 0.322i)11-s + 0.573·13-s + (−0.121 − 0.696i)14-s + (−0.125 + 0.216i)16-s + (−0.482 − 0.835i)17-s + (0.888 − 1.53i)19-s + 0.675·20-s − 0.263·22-s + (0.357 − 0.619i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.283 + 0.958i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.283 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.283 + 0.958i$
Analytic conductor: \(20.2083\)
Root analytic conductor: \(4.49537\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :5/2),\ -0.283 + 0.958i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.774437489\)
\(L(\frac12)\) \(\approx\) \(1.774437489\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2 + 3.46i)T \)
3 \( 1 \)
7 \( 1 + (-99.4 + 83.1i)T \)
good5 \( 1 + (37.7 - 65.3i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (74.7 + 129. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 - 349.T + 3.71e5T^{2} \)
17 \( 1 + (574. + 995. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-1.39e3 + 2.42e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-906. + 1.57e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 - 759.T + 2.05e7T^{2} \)
31 \( 1 + (4.51e3 + 7.82e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (3.89e3 - 6.75e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + 7.64e3T + 1.15e8T^{2} \)
43 \( 1 - 1.21e4T + 1.47e8T^{2} \)
47 \( 1 + (-1.22e4 + 2.13e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-6.79e3 - 1.17e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (1.31e4 + 2.28e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (1.76e4 - 3.05e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (2.71e4 + 4.70e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 7.01e4T + 1.80e9T^{2} \)
73 \( 1 + (-2.22e4 - 3.85e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (3.08e4 - 5.33e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 8.71e4T + 3.93e9T^{2} \)
89 \( 1 + (-4.92e4 + 8.53e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 3.23e4T + 8.58e9T^{2} \)
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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

−11.75177074711858353578277098089, −11.15636219001585365005270904404, −10.54997256087262906756088590123, −9.074350102716754245921414724597, −7.65066742977170227177663064754, −6.73206824287478074101875780849, −5.02111176747107316413682331182, −3.76025057644362189423688096054, −2.58068510479999918438822279089, −0.61348372369356748237067130245, 1.46635481624463237676080239482, 3.74275675750012345361824928925, 4.91503309142647855348890016267, 5.81834115349266830926858240154, 7.54417407631467050454184774319, 8.370005846194992190988347012092, 9.146089291219129019229596722986, 10.86875413472673608775950323597, 12.16731854174008987027063596343, 12.53446872502082297497047132326

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