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Properties

Label 2-2312-136.45-c0-0-1
Degree $2$
Conductor $2312$
Sign $0.547 - 0.837i$
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 + (1.38 − 0.275i)3-s + (0.707 + 0.707i)4-s + (−0.785 + 1.17i)5-s + (1.38 + 0.275i)6-s + (0.382 + 0.923i)8-s + (0.923 − 0.382i)9-s + (−1.17 + 0.785i)10-s + (0.275 − 1.38i)11-s + (1.17 + 0.785i)12-s + (−0.765 + 1.84i)15-s + i·16-s + 1.00·18-s + (−1.38 + 0.275i)20-s + (0.785 − 1.17i)22-s + ⋯
L(s)  = 1  + (0.923 + 0.382i)2-s + (1.38 − 0.275i)3-s + (0.707 + 0.707i)4-s + (−0.785 + 1.17i)5-s + (1.38 + 0.275i)6-s + (0.382 + 0.923i)8-s + (0.923 − 0.382i)9-s + (−1.17 + 0.785i)10-s + (0.275 − 1.38i)11-s + (1.17 + 0.785i)12-s + (−0.765 + 1.84i)15-s + i·16-s + 1.00·18-s + (−1.38 + 0.275i)20-s + (0.785 − 1.17i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.547 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.547 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2312\)    =    \(2^{3} \cdot 17^{2}\)
Sign: $0.547 - 0.837i$
Analytic conductor: \(1.15383\)
Root analytic conductor: \(1.07416\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2312} (1405, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2312,\ (\ :0),\ 0.547 - 0.837i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.845572110\)
\(L(\frac12)\) \(\approx\) \(2.845572110\)
\(L(1)\) 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 \( 1 + (-1.38 + 0.275i)T + (0.923 - 0.382i)T^{2} \)
5 \( 1 + (0.785 - 1.17i)T + (-0.382 - 0.923i)T^{2} \)
7 \( 1 + (0.382 - 0.923i)T^{2} \)
11 \( 1 + (-0.275 + 1.38i)T + (-0.923 - 0.382i)T^{2} \)
13 \( 1 + iT^{2} \)
19 \( 1 + (-0.707 - 0.707i)T^{2} \)
23 \( 1 + (-0.923 - 0.382i)T^{2} \)
29 \( 1 + (1.17 + 0.785i)T + (0.382 + 0.923i)T^{2} \)
31 \( 1 + (0.923 - 0.382i)T^{2} \)
37 \( 1 + (1.38 - 0.275i)T + (0.923 - 0.382i)T^{2} \)
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 \( 1 + (-1.17 + 0.785i)T + (0.382 - 0.923i)T^{2} \)
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

−8.858769202191014142917644081087, −8.395795349159735025363275359794, −7.60009980523225122244629768823, −7.16269834583784793706659218295, −6.32768221695092757116464751478, −5.46870975694230385759035698765, −4.03965368846726340598409227446, −3.50619578552808721015676803979, −2.98439171005581093183528712837, −2.06267135061737335327847237330, 1.50343849721579655396299460629, 2.39483197300970969054527950053, 3.54812953124466308327035933229, 4.08810887385244228912938017361, 4.75405917199111511998477663953, 5.56057235588183739790956109035, 6.98391487141189823050458809480, 7.50630463008640665871622216783, 8.392486151387196027941234912024, 9.124695441493975341988626598281

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