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Properties

Label 2-2312-136.115-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 + (1 + i)3-s − 4-s + (1 − i)6-s + i·8-s + i·9-s + (−1 + i)11-s + (−1 − i)12-s + 16-s + 18-s + 2i·19-s + (1 + i)22-s + (−1 + i)24-s + i·25-s − i·32-s − 2·33-s + ⋯
L(s)  = 1  − i·2-s + (1 + i)3-s − 4-s + (1 − i)6-s + i·8-s + i·9-s + (−1 + i)11-s + (−1 − i)12-s + 16-s + 18-s + 2i·19-s + (1 + i)22-s + (−1 + i)24-s + i·25-s − i·32-s − 2·33-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: $\chi_{2312} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2312,\ (\ :0),\ 0.788 - 0.615i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.321992698\)
\(L(\frac12)\) \(\approx\) \(1.321992698\)
\(L(1)\) 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 \( 1 + (-1 - i)T + iT^{2} \)
5 \( 1 - iT^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + (1 - i)T - iT^{2} \)
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 \( 1 + (-1 + i)T - iT^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + iT^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 + (-1 - i)T + iT^{2} \)
79 \( 1 + iT^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (1 + i)T + iT^{2} \)
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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

−9.578716163200738834477334255141, −8.660174787931424491343480036435, −8.075760031026605745351206774004, −7.33456217188006719694955678008, −5.74100164481167738636768386791, −5.01251587117738563151859986688, −4.10674858456214283759283665622, −3.56474124342709356998305094179, −2.62730330798888613834246385901, −1.77402779539533271587330371493, 0.821569623257130770884288497517, 2.49843459899612704812075948417, 3.14631737396415544365951699766, 4.43584965500702648356072977155, 5.28861005223960289897663629154, 6.28203709917504211777574357983, 6.86612811204264592196064233154, 7.76950220363082772312762822276, 8.061037241949871218661066524175, 8.879878327375682117319806791480

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