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Properties

Label 2-2312-136.125-c0-0-1
Degree $2$
Conductor $2312$
Sign $0.507 + 0.861i$
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  + (−0.923 − 0.382i)2-s + (0.275 + 1.38i)3-s + (0.707 + 0.707i)4-s + (−1.17 − 0.785i)5-s + (0.275 − 1.38i)6-s + (−0.382 − 0.923i)8-s + (−0.923 + 0.382i)9-s + (0.785 + 1.17i)10-s + (−1.38 − 0.275i)11-s + (−0.785 + 1.17i)12-s + (0.765 − 1.84i)15-s + i·16-s + 1.00·18-s + (−0.275 − 1.38i)20-s + (1.17 + 0.785i)22-s + ⋯
L(s)  = 1  + (−0.923 − 0.382i)2-s + (0.275 + 1.38i)3-s + (0.707 + 0.707i)4-s + (−1.17 − 0.785i)5-s + (0.275 − 1.38i)6-s + (−0.382 − 0.923i)8-s + (−0.923 + 0.382i)9-s + (0.785 + 1.17i)10-s + (−1.38 − 0.275i)11-s + (−0.785 + 1.17i)12-s + (0.765 − 1.84i)15-s + i·16-s + 1.00·18-s + (−0.275 − 1.38i)20-s + (1.17 + 0.785i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2312\)    =    \(2^{3} \cdot 17^{2}\)
Sign: $0.507 + 0.861i$
Analytic conductor: \(1.15383\)
Root analytic conductor: \(1.07416\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2312} (1485, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2312,\ (\ :0),\ 0.507 + 0.861i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4347400925\)
\(L(\frac12)\) \(\approx\) \(0.4347400925\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.923 + 0.382i)T \)
17 \( 1 \)
good3 \( 1 + (-0.275 - 1.38i)T + (-0.923 + 0.382i)T^{2} \)
5 \( 1 + (1.17 + 0.785i)T + (0.382 + 0.923i)T^{2} \)
7 \( 1 + (-0.382 + 0.923i)T^{2} \)
11 \( 1 + (1.38 + 0.275i)T + (0.923 + 0.382i)T^{2} \)
13 \( 1 + iT^{2} \)
19 \( 1 + (-0.707 - 0.707i)T^{2} \)
23 \( 1 + (0.923 + 0.382i)T^{2} \)
29 \( 1 + (-0.785 + 1.17i)T + (-0.382 - 0.923i)T^{2} \)
31 \( 1 + (-0.923 + 0.382i)T^{2} \)
37 \( 1 + (0.275 + 1.38i)T + (-0.923 + 0.382i)T^{2} \)
41 \( 1 + (-0.382 + 0.923i)T^{2} \)
43 \( 1 + (-0.707 + 0.707i)T^{2} \)
47 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
53 \( 1 + (-0.707 - 0.707i)T^{2} \)
59 \( 1 + (0.707 - 0.707i)T^{2} \)
61 \( 1 + (0.785 + 1.17i)T + (-0.382 + 0.923i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (0.923 - 0.382i)T^{2} \)
73 \( 1 + (-0.382 - 0.923i)T^{2} \)
79 \( 1 + (-0.923 - 0.382i)T^{2} \)
83 \( 1 + (0.707 + 0.707i)T^{2} \)
89 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
97 \( 1 + (0.382 + 0.923i)T^{2} \)
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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

−8.957652283141445327831952210957, −8.511903756611037912258543139519, −7.88607936325822863364008797466, −7.18301821205449083818205265433, −5.74734706328323891622765918081, −4.81103752156142559649368934521, −4.05321340063120015899792197832, −3.38220674620865852254503576588, −2.36860130935488648500675306694, −0.42433918697253899866287740568, 1.19834267718127877410008716630, 2.51273377376037941098579267736, 3.04738405427315310166319338727, 4.64064690578712525417103658741, 5.78525677207445607826970412040, 6.68656057012786685616136826039, 7.24189581351330545634869769063, 7.72268322219571038971110303666, 8.198507641264859888460184393117, 8.957914050181407846721429729486

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