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Properties

Label 2-2312-136.59-c0-0-0
Degree $2$
Conductor $2312$
Sign $-0.0465 - 0.998i$
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 − 1.00i·4-s + (0.707 + 0.707i)8-s + (0.707 + 0.707i)9-s − 1.00·16-s − 1.00·18-s + (−1.41 + 1.41i)19-s + (0.707 + 0.707i)25-s + (0.707 − 0.707i)32-s + (0.707 − 0.707i)36-s − 2.00i·38-s + (1.41 + 1.41i)43-s + (0.707 − 0.707i)49-s − 1.00·50-s + (−1.41 − 1.41i)59-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (0.707 + 0.707i)8-s + (0.707 + 0.707i)9-s − 1.00·16-s − 1.00·18-s + (−1.41 + 1.41i)19-s + (0.707 + 0.707i)25-s + (0.707 − 0.707i)32-s + (0.707 − 0.707i)36-s − 2.00i·38-s + (1.41 + 1.41i)43-s + (0.707 − 0.707i)49-s − 1.00·50-s + (−1.41 − 1.41i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0465 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0465 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2312\)    =    \(2^{3} \cdot 17^{2}\)
Sign: $-0.0465 - 0.998i$
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: \((2,\ 2312,\ (\ :0),\ -0.0465 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8035723939\)
\(L(\frac12)\) \(\approx\) \(0.8035723939\)
\(L(1)\) 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 \( 1 + (-0.707 - 0.707i)T^{2} \)
5 \( 1 + (-0.707 - 0.707i)T^{2} \)
7 \( 1 + (-0.707 + 0.707i)T^{2} \)
11 \( 1 + (-0.707 + 0.707i)T^{2} \)
13 \( 1 + T^{2} \)
19 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
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 \( 1 + (0.707 - 0.707i)T^{2} \)
43 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
61 \( 1 + (-0.707 + 0.707i)T^{2} \)
67 \( 1 - 2T + T^{2} \)
71 \( 1 + (0.707 + 0.707i)T^{2} \)
73 \( 1 + (0.707 + 0.707i)T^{2} \)
79 \( 1 + (0.707 - 0.707i)T^{2} \)
83 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
89 \( 1 - 2iT - T^{2} \)
97 \( 1 + (0.707 + 0.707i)T^{2} \)
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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

−9.403926463612349682862802896759, −8.381102851067254328076400921273, −7.978750205320034444318234199382, −7.13530153199648291528706814521, −6.43433094806928459563688170687, −5.61462328686887477443063831466, −4.76431076193265845511270898639, −3.92104226276167143644047942918, −2.33974253115638221557235753819, −1.38493904020624041157621333140, 0.75754313721658315546386503779, 2.09267031624450672114732513965, 2.97839552388439962333696984792, 4.10785915539618027550561037447, 4.60894539258318054103948314037, 6.07559292753882842868518072876, 6.92910225242056511926518005962, 7.42765568728305929055028984287, 8.616406707242674613786241759631, 8.920688990063562494784673607256

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