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Properties

Label 2-2312-136.59-c0-0-4
Degree $2$
Conductor $2312$
Sign $-0.815 + 0.578i$
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  + (0.707 − 0.707i)2-s + (0.292 − 0.707i)3-s − 1.00i·4-s + (−0.292 − 0.707i)6-s + (−0.707 − 0.707i)8-s + (0.292 + 0.292i)9-s + (−0.707 − 1.70i)11-s + (−0.707 − 0.292i)12-s − 1.00·16-s + 0.414·18-s + (−1.70 − 0.707i)22-s + (−0.707 + 0.292i)24-s + (0.707 + 0.707i)25-s + (1.00 − 0.414i)27-s + (−0.707 + 0.707i)32-s − 1.41·33-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s + (0.292 − 0.707i)3-s − 1.00i·4-s + (−0.292 − 0.707i)6-s + (−0.707 − 0.707i)8-s + (0.292 + 0.292i)9-s + (−0.707 − 1.70i)11-s + (−0.707 − 0.292i)12-s − 1.00·16-s + 0.414·18-s + (−1.70 − 0.707i)22-s + (−0.707 + 0.292i)24-s + (0.707 + 0.707i)25-s + (1.00 − 0.414i)27-s + (−0.707 + 0.707i)32-s − 1.41·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.815 + 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.815 + 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2312\)    =    \(2^{3} \cdot 17^{2}\)
Sign: $-0.815 + 0.578i$
Analytic conductor: \(1.15383\)
Root analytic conductor: \(1.07416\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2312} (1555, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2312,\ (\ :0),\ -0.815 + 0.578i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.787558798\)
\(L(\frac12)\) \(\approx\) \(1.787558798\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
17 \( 1 \)
good3 \( 1 + (-0.292 + 0.707i)T + (-0.707 - 0.707i)T^{2} \)
5 \( 1 + (-0.707 - 0.707i)T^{2} \)
7 \( 1 + (-0.707 + 0.707i)T^{2} \)
11 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
13 \( 1 + T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + (0.707 - 0.707i)T^{2} \)
29 \( 1 + (-0.707 - 0.707i)T^{2} \)
31 \( 1 + (0.707 + 0.707i)T^{2} \)
37 \( 1 + (0.707 + 0.707i)T^{2} \)
41 \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \)
43 \( 1 + (1 + i)T + iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (-1 - i)T + iT^{2} \)
61 \( 1 + (-0.707 + 0.707i)T^{2} \)
67 \( 1 - 1.41T + T^{2} \)
71 \( 1 + (0.707 + 0.707i)T^{2} \)
73 \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \)
79 \( 1 + (0.707 - 0.707i)T^{2} \)
83 \( 1 + (1 - i)T - iT^{2} \)
89 \( 1 + 1.41iT - T^{2} \)
97 \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \)
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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

−8.681163341982505411082755690833, −8.313098019841603258510859991378, −7.16967909158563506326442761100, −6.52818499746626937673798341923, −5.52391095157483572058644915441, −5.00884317960358605439387161079, −3.74122271426789512552308595914, −3.02017203267256526924144619748, −2.11104321911013242560180994210, −0.964026774247350729469339439108, 2.10763728848378694077928224518, 3.14675575091172041872282584900, 4.05352148895143400870790120685, 4.76688213346686164127109183059, 5.26119811898746851056005566472, 6.58241470395438519739889163446, 6.98645235187869563836299505892, 7.903462896515210515092950206772, 8.596517998296549550665956250203, 9.519063193044402835460606359081

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