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Properties

Label 2-2312-136.123-c0-0-1
Degree $2$
Conductor $2312$
Sign $-0.788 - 0.615i$
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  + i·2-s − 4-s − i·8-s + i·9-s + 16-s − 18-s + 2i·19-s − i·25-s + i·32-s − i·36-s − 2·38-s + 2i·43-s + i·49-s + 50-s + 2i·59-s + ⋯
L(s)  = 1  + i·2-s − 4-s − i·8-s + i·9-s + 16-s − 18-s + 2i·19-s − i·25-s + i·32-s − i·36-s − 2·38-s + 2i·43-s + i·49-s + 50-s + 2i·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2312\)    =    \(2^{3} \cdot 17^{2}\)
Sign: $-0.788 - 0.615i$
Analytic conductor: \(1.15383\)
Root analytic conductor: \(1.07416\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2312} (1483, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2312,\ (\ :0),\ -0.788 - 0.615i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9259394931\)
\(L(\frac12)\) \(\approx\) \(0.9259394931\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.335611276163349758192664787828, −8.466860694557162675314857351117, −7.86081569489897969825483506096, −7.38804052050323271879514453560, −6.18590656082949613562344171050, −5.83262475303969927597826677481, −4.75995162467162920606528561261, −4.16600249811789502870141340453, −2.98734769845079586076909409057, −1.54251358439799015608499044872, 0.67058649002567007435868247561, 2.03028937566543195437490777910, 3.08569167941142029018355060881, 3.78376855625802107473895528507, 4.77538107796779181654585196364, 5.49400919323477587943150944247, 6.60237865178392543933683060941, 7.34118638870927065654233743289, 8.469556659940864930462060267162, 9.109161792543052050113103285253

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