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Properties

Label 2-2912-8.5-c1-0-54
Degree $2$
Conductor $2912$
Sign $0.485 + 0.874i$
Analytic cond. $23.2524$
Root an. cond. $4.82207$
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  + 0.302i·3-s − 1.42i·5-s + 7-s + 2.90·9-s − 2.11i·11-s + i·13-s + 0.431·15-s + 1.07·17-s − 3.52i·19-s + 0.302i·21-s − 5.47·23-s + 2.95·25-s + 1.78i·27-s − 3.52i·29-s + 3.26·31-s + ⋯
L(s)  = 1  + 0.174i·3-s − 0.638i·5-s + 0.377·7-s + 0.969·9-s − 0.636i·11-s + 0.277i·13-s + 0.111·15-s + 0.260·17-s − 0.808i·19-s + 0.0659i·21-s − 1.14·23-s + 0.591·25-s + 0.343i·27-s − 0.655i·29-s + 0.585·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.485 + 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.485 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2912\)    =    \(2^{5} \cdot 7 \cdot 13\)
Sign: $0.485 + 0.874i$
Analytic conductor: \(23.2524\)
Root analytic conductor: \(4.82207\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2912} (1457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2912,\ (\ :1/2),\ 0.485 + 0.874i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.004260377\)
\(L(\frac12)\) \(\approx\) \(2.004260377\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - T \)
13 \( 1 - iT \)
good3 \( 1 - 0.302iT - 3T^{2} \)
5 \( 1 + 1.42iT - 5T^{2} \)
11 \( 1 + 2.11iT - 11T^{2} \)
17 \( 1 - 1.07T + 17T^{2} \)
19 \( 1 + 3.52iT - 19T^{2} \)
23 \( 1 + 5.47T + 23T^{2} \)
29 \( 1 + 3.52iT - 29T^{2} \)
31 \( 1 - 3.26T + 31T^{2} \)
37 \( 1 + 10.7iT - 37T^{2} \)
41 \( 1 - 9.67T + 41T^{2} \)
43 \( 1 - 9.02iT - 43T^{2} \)
47 \( 1 + 4.79T + 47T^{2} \)
53 \( 1 - 0.906iT - 53T^{2} \)
59 \( 1 + 6.88iT - 59T^{2} \)
61 \( 1 - 5.18iT - 61T^{2} \)
67 \( 1 - 5.46iT - 67T^{2} \)
71 \( 1 + 0.895T + 71T^{2} \)
73 \( 1 + 14.4T + 73T^{2} \)
79 \( 1 + 15.5T + 79T^{2} \)
83 \( 1 + 9.02iT - 83T^{2} \)
89 \( 1 - 17.4T + 89T^{2} \)
97 \( 1 - 4.26T + 97T^{2} \)
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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

−8.692761469955547399438977123151, −7.88823255732086070753451594962, −7.26109190094529282527197365338, −6.26713099568238755187602458594, −5.54729860069654963314210836972, −4.50807785708074038680856169638, −4.19842691693804675974265714179, −2.93589313078025810899015870871, −1.77410955643068577978395162802, −0.70580524097806115019026305704, 1.24817423986762065412165236371, 2.17690475088846209768517715707, 3.26183714606462721013575831511, 4.19758172805504599725461971195, 4.92544163074904767055554785673, 5.96902246869458252077790044072, 6.69358402132175575584261328778, 7.42439000506224329594560371513, 7.949585562425278905075820591180, 8.817717556627850381148366819168

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