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Properties

Label 2-2312-136.67-c0-0-3
Degree $2$
Conductor $2312$
Sign $-0.727 + 0.685i$
Analytic cond. $1.15383$
Root an. cond. $1.07416$
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  − 2-s − 1.41i·3-s + 4-s + 1.41i·6-s − 8-s − 1.00·9-s − 1.41i·11-s − 1.41i·12-s + 16-s + 1.00·18-s + 2·19-s + 1.41i·22-s + 1.41i·24-s − 25-s − 32-s − 2.00·33-s + ⋯
L(s)  = 1  − 2-s − 1.41i·3-s + 4-s + 1.41i·6-s − 8-s − 1.00·9-s − 1.41i·11-s − 1.41i·12-s + 16-s + 1.00·18-s + 2·19-s + 1.41i·22-s + 1.41i·24-s − 25-s − 32-s − 2.00·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.727 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.727 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2312\)    =    \(2^{3} \cdot 17^{2}\)
Sign: $-0.727 + 0.685i$
Analytic conductor: \(1.15383\)
Root analytic conductor: \(1.07416\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2312} (1155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2312,\ (\ :0),\ -0.727 + 0.685i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7253288398\)
\(L(\frac12)\) \(\approx\) \(0.7253288398\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
17 \( 1 \)
good3 \( 1 + 1.41iT - T^{2} \)
5 \( 1 + T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + 1.41iT - T^{2} \)
13 \( 1 - T^{2} \)
19 \( 1 - 2T + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - 1.41iT - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + 1.41iT - T^{2} \)
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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

−8.707458078458520920068480075152, −8.031326241540536464289443557890, −7.48956473704374004536494950397, −6.83830519716989257265969354464, −5.97865340789028463133336847762, −5.46389556050631479372512015207, −3.57168116561604662871776414935, −2.74398596629721884571899181962, −1.64456328964810622575787843166, −0.72067421084153552208929433333, 1.58160263771941075541406582282, 2.85772248277175471962597874792, 3.71682493019203716489889266243, 4.74657607714187197422294611105, 5.43953426940813327881853049335, 6.49323256411014365129778835052, 7.45270596970687120752617307169, 7.953517020825544983906502758866, 9.102005453951643520433010899772, 9.614264772597664196256758051938

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