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Properties

Label 2-2312-136.115-c0-0-2
Degree $2$
Conductor $2312$
Sign $-0.999 - 0.0124i$
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 + (1.30 + 1.30i)3-s − 4-s + (−1.30 + 1.30i)6-s − i·8-s + 2.41i·9-s + (−0.541 + 0.541i)11-s + (−1.30 − 1.30i)12-s + 16-s − 2.41·18-s + (−0.541 − 0.541i)22-s + (1.30 − 1.30i)24-s + i·25-s + (−1.84 + 1.84i)27-s + i·32-s − 1.41·33-s + ⋯
L(s)  = 1  + i·2-s + (1.30 + 1.30i)3-s − 4-s + (−1.30 + 1.30i)6-s − i·8-s + 2.41i·9-s + (−0.541 + 0.541i)11-s + (−1.30 − 1.30i)12-s + 16-s − 2.41·18-s + (−0.541 − 0.541i)22-s + (1.30 − 1.30i)24-s + i·25-s + (−1.84 + 1.84i)27-s + i·32-s − 1.41·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0124i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0124i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2312\)    =    \(2^{3} \cdot 17^{2}\)
Sign: $-0.999 - 0.0124i$
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: \((2,\ 2312,\ (\ :0),\ -0.999 - 0.0124i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.530020124\)
\(L(\frac12)\) \(\approx\) \(1.530020124\)
\(L(1)\) 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 \( 1 + (-1.30 - 1.30i)T + iT^{2} \)
5 \( 1 - iT^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + (0.541 - 0.541i)T - iT^{2} \)
13 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (0.541 - 0.541i)T - iT^{2} \)
43 \( 1 + 1.41iT - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + 1.41iT - T^{2} \)
61 \( 1 + iT^{2} \)
67 \( 1 - 1.41T + T^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 + (-0.541 - 0.541i)T + iT^{2} \)
79 \( 1 + iT^{2} \)
83 \( 1 + 1.41iT - T^{2} \)
89 \( 1 - 1.41T + T^{2} \)
97 \( 1 + (-1.30 - 1.30i)T + iT^{2} \)
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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

−9.301537793510934717371211056326, −8.864810867888263971498066198391, −8.042765999682718482911452897786, −7.56378795872662437041138517522, −6.60827130282825585942485685934, −5.25531452446325118790529756575, −4.96093372171695306129004221416, −3.91041145608858048876225568197, −3.35807395878694034466747875955, −2.13315064029671857441814791339, 0.927467098985324896442934865830, 2.03474123137513692719430069722, 2.76874431148807665957516071096, 3.44866184685098969778769136389, 4.46631463829801752996923246455, 5.72278876903946475125999652953, 6.60507509979479617220373296874, 7.62897063622446631441577595884, 8.157901485652754734334199232637, 8.738240012030327074391859074335

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