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Properties

Label 2-2312-136.19-c0-0-3
Degree $2$
Conductor $2312$
Sign $-0.758 + 0.651i$
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 + (−1.70 + 0.707i)3-s + 1.00i·4-s + (1.70 + 0.707i)6-s + (0.707 − 0.707i)8-s + (1.70 − 1.70i)9-s + (−0.707 − 0.292i)11-s + (−0.707 − 1.70i)12-s − 1.00·16-s − 2.41·18-s + (0.292 + 0.707i)22-s + (−0.707 + 1.70i)24-s + (−0.707 + 0.707i)25-s + (−1 + 2.41i)27-s + (0.707 + 0.707i)32-s + 1.41·33-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + (−1.70 + 0.707i)3-s + 1.00i·4-s + (1.70 + 0.707i)6-s + (0.707 − 0.707i)8-s + (1.70 − 1.70i)9-s + (−0.707 − 0.292i)11-s + (−0.707 − 1.70i)12-s − 1.00·16-s − 2.41·18-s + (0.292 + 0.707i)22-s + (−0.707 + 1.70i)24-s + (−0.707 + 0.707i)25-s + (−1 + 2.41i)27-s + (0.707 + 0.707i)32-s + 1.41·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.758 + 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.758 + 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2312\)    =    \(2^{3} \cdot 17^{2}\)
Sign: $-0.758 + 0.651i$
Analytic conductor: \(1.15383\)
Root analytic conductor: \(1.07416\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2312} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2312,\ (\ :0),\ -0.758 + 0.651i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1512170640\)
\(L(\frac12)\) \(\approx\) \(0.1512170640\)
\(L(1)\) 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 \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \)
5 \( 1 + (0.707 - 0.707i)T^{2} \)
7 \( 1 + (0.707 + 0.707i)T^{2} \)
11 \( 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2} \)
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 \( 1 + (-0.292 + 0.707i)T + (-0.707 - 0.707i)T^{2} \)
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 \( 1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2} \)
79 \( 1 + (-0.707 - 0.707i)T^{2} \)
83 \( 1 + (1 + i)T + iT^{2} \)
89 \( 1 + 1.41iT - T^{2} \)
97 \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \)
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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

−9.247501416684215245842310000924, −8.229821177271143368542127408306, −7.34187622376694061997087510516, −6.53093375642412423827995305794, −5.64900797737924428192190244863, −4.89610692550134970422769839245, −4.05667128789788343886601695066, −3.14044592301142469205141518119, −1.61092849066624977269920437601, −0.18513576429461054648880901495, 1.19679698978785781686125433133, 2.29133800822006326032054308846, 4.35339893964438940697049711091, 5.13346346501427914224441101816, 5.75908470139773285013347766207, 6.41606049100249994401689411073, 7.11024702636356563274202752869, 7.72389298271732695549682760900, 8.451503125613372270536638346863, 9.656190222277760248183657315493

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