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Properties

Label 2-2912-728.51-c0-0-3
Degree $2$
Conductor $2912$
Sign $-0.592 + 0.805i$
Analytic cond. $1.45327$
Root an. cond. $1.20551$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.939 − 1.62i)3-s + (−0.173 + 0.300i)5-s + (0.939 − 0.342i)7-s + (−1.26 + 2.19i)9-s + 13-s + 0.652·15-s + (−0.766 − 1.32i)17-s + (−1.43 − 1.20i)21-s + (0.439 + 0.761i)25-s + 2.87·27-s + (−0.5 − 0.866i)31-s + (−0.0603 + 0.342i)35-s + (0.939 − 1.62i)37-s + (−0.939 − 1.62i)39-s − 0.347·43-s + ⋯
L(s)  = 1  + (−0.939 − 1.62i)3-s + (−0.173 + 0.300i)5-s + (0.939 − 0.342i)7-s + (−1.26 + 2.19i)9-s + 13-s + 0.652·15-s + (−0.766 − 1.32i)17-s + (−1.43 − 1.20i)21-s + (0.439 + 0.761i)25-s + 2.87·27-s + (−0.5 − 0.866i)31-s + (−0.0603 + 0.342i)35-s + (0.939 − 1.62i)37-s + (−0.939 − 1.62i)39-s − 0.347·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2912\)    =    \(2^{5} \cdot 7 \cdot 13\)
Sign: $-0.592 + 0.805i$
Analytic conductor: \(1.45327\)
Root analytic conductor: \(1.20551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2912} (1871, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2912,\ (\ :0),\ -0.592 + 0.805i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8829806419\)
\(L(\frac12)\) \(\approx\) \(0.8829806419\)
\(L(1)\) 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 \( 1 + (-0.939 + 0.342i)T \)
13 \( 1 - T \)
good3 \( 1 + (0.939 + 1.62i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 0.347T + T^{2} \)
47 \( 1 + (-0.766 + 1.32i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + 1.53T + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.453931892045779256912304649856, −7.55778838504746284147080619599, −7.28359127427842327190374622957, −6.53145787190464579358688306184, −5.70248027103083624734694541947, −5.09230149151264956807245671893, −4.03306674595143819329618201619, −2.60813367613237543960410316128, −1.74071724294216539730130331108, −0.70066610543839740004695365543, 1.35085236258982846614393319354, 2.98739050687758346227149299459, 4.09687123705724491273729275554, 4.44742862178986267091640556491, 5.24114537335889273936726588300, 5.99689163070152377415474535956, 6.55322136402297821022327294312, 8.038972168187291765764862193090, 8.665340244582914457324717439475, 9.115852557113457270685953190876

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