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Properties

Label 2-912-57.50-c1-0-23
Degree $2$
Conductor $912$
Sign $0.823 - 0.567i$
Analytic cond. $7.28235$
Root an. cond. $2.69858$
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  + (0.5 + 1.65i)3-s + (3.68 − 2.12i)5-s + 2.37·7-s + (−2.5 + 1.65i)9-s + 1.58i·11-s + (1.5 + 0.866i)13-s + (5.37 + 5.04i)15-s + (−2.31 + 1.33i)17-s + (−4 + 1.73i)19-s + (1.18 + 3.93i)21-s + (3.68 + 2.12i)23-s + (6.55 − 11.3i)25-s + (−4 − 3.31i)27-s + (2.31 − 4.00i)29-s − 7.57i·31-s + ⋯
L(s)  = 1  + (0.288 + 0.957i)3-s + (1.64 − 0.951i)5-s + 0.896·7-s + (−0.833 + 0.552i)9-s + 0.477i·11-s + (0.416 + 0.240i)13-s + (1.38 + 1.30i)15-s + (−0.561 + 0.324i)17-s + (−0.917 + 0.397i)19-s + (0.258 + 0.858i)21-s + (0.768 + 0.443i)23-s + (1.31 − 2.27i)25-s + (−0.769 − 0.638i)27-s + (0.429 − 0.744i)29-s − 1.36i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 - 0.567i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.823 - 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.823 - 0.567i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.823 - 0.567i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.33897 + 0.728129i\)
\(L(\frac12)\) \(\approx\) \(2.33897 + 0.728129i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.5 - 1.65i)T \)
19 \( 1 + (4 - 1.73i)T \)
good5 \( 1 + (-3.68 + 2.12i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 2.37T + 7T^{2} \)
11 \( 1 - 1.58iT - 11T^{2} \)
13 \( 1 + (-1.5 - 0.866i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.31 - 1.33i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-3.68 - 2.12i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.31 + 4.00i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.57iT - 31T^{2} \)
37 \( 1 - 4.10iT - 37T^{2} \)
41 \( 1 + (-5.05 - 8.76i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.87 + 8.43i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.31 + 1.33i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.68 - 6.38i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.68 - 6.38i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.87 + 8.43i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.5 - 0.866i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.05 + 8.76i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.12 + 1.95i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.61 - 4.97i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 8.51iT - 83T^{2} \)
89 \( 1 + (3.68 - 6.38i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (15.1 - 8.76i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.962758128023690569285551803848, −9.437293228733763952557494002355, −8.639525552171937475808914969476, −8.081201558070008948597765060360, −6.44412889087562354106942752003, −5.65081050990467645985753161368, −4.80990970981863814103446228049, −4.21245870042591326229214159312, −2.47180183643775377772621288990, −1.59009123824527270814174716050, 1.38869151938880182520436965990, 2.33090277534842738108762076190, 3.16041211862848878271416056519, 4.97648360924277591465545911459, 5.88824628047210110311841928379, 6.64348530397191925355040092686, 7.21107808252434384312644589584, 8.546062009703249848209939874504, 8.906953346550518044410523335117, 10.09686895036051356693453235994

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