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Properties

Label 2-2312-136.43-c0-0-5
Degree $2$
Conductor $2312$
Sign $0.0340 - 0.999i$
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 + (−1.30 − 0.541i)3-s − 1.00i·4-s + (−1.30 + 0.541i)6-s + (−0.707 − 0.707i)8-s + (0.707 + 0.707i)9-s + (−1.30 + 0.541i)11-s + (−0.541 + 1.30i)12-s − 1.00·16-s + 1.00·18-s + (−1.41 + 1.41i)19-s + (−0.541 + 1.30i)22-s + (0.541 + 1.30i)24-s + (−0.707 − 0.707i)25-s + (−0.707 + 0.707i)32-s + 2·33-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s + (−1.30 − 0.541i)3-s − 1.00i·4-s + (−1.30 + 0.541i)6-s + (−0.707 − 0.707i)8-s + (0.707 + 0.707i)9-s + (−1.30 + 0.541i)11-s + (−0.541 + 1.30i)12-s − 1.00·16-s + 1.00·18-s + (−1.41 + 1.41i)19-s + (−0.541 + 1.30i)22-s + (0.541 + 1.30i)24-s + (−0.707 − 0.707i)25-s + (−0.707 + 0.707i)32-s + 2·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0340 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0340 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2312\)    =    \(2^{3} \cdot 17^{2}\)
Sign: $0.0340 - 0.999i$
Analytic conductor: \(1.15383\)
Root analytic conductor: \(1.07416\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2312} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2312,\ (\ :0),\ 0.0340 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01917871715\)
\(L(\frac12)\) \(\approx\) \(0.01917871715\)
\(L(1)\) 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 \( 1 + (1.30 + 0.541i)T + (0.707 + 0.707i)T^{2} \)
5 \( 1 + (0.707 + 0.707i)T^{2} \)
7 \( 1 + (0.707 - 0.707i)T^{2} \)
11 \( 1 + (1.30 - 0.541i)T + (0.707 - 0.707i)T^{2} \)
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 \( 1 + (0.541 + 1.30i)T + (-0.707 + 0.707i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (0.707 - 0.707i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (-0.707 - 0.707i)T^{2} \)
73 \( 1 + (0.541 - 1.30i)T + (-0.707 - 0.707i)T^{2} \)
79 \( 1 + (-0.707 + 0.707i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.541 - 1.30i)T + (-0.707 - 0.707i)T^{2} \)
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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

−8.582051775693450735431189984296, −7.66092736527158021305948694791, −6.73672510411999652078575039144, −6.02392477784236726347682157182, −5.48469666911059350348035552594, −4.69812673536794768578245813718, −3.84233502256158039328740147862, −2.49208525649374101695453126519, −1.62388067802965991236320491340, −0.01135336957170926200202113786, 2.43862206839291801959350529122, 3.50559486597387139909327336238, 4.67265521073079091434367388678, 4.95497736745412027062781900266, 5.85461392056205052551439665256, 6.35930399238639030035931046177, 7.23053804079752615670303589175, 8.112694572185609098480151016123, 8.823031235493531295420680526626, 9.883702062746596924560779573303

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