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Properties

Label 2-3104-776.315-c0-0-0
Degree $2$
Conductor $3104$
Sign $-0.634 - 0.773i$
Analytic cond. $1.54909$
Root an. cond. $1.24462$
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  + (0.5 + 1.86i)3-s + (−2.36 + 1.36i)9-s + (1.67 + 0.448i)11-s + (0.0999 + 0.758i)17-s + (1.70 + 0.707i)19-s + (0.258 − 0.965i)25-s + (−2.36 − 2.36i)27-s + 3.34i·33-s + (−1.12 − 1.46i)41-s + (−0.448 − 0.258i)43-s + (−0.965 − 0.258i)49-s + (−1.36 + 0.565i)51-s + (−0.465 + 3.53i)57-s + (−0.758 − 0.0999i)59-s + (−1.12 − 0.465i)67-s + ⋯
L(s)  = 1  + (0.5 + 1.86i)3-s + (−2.36 + 1.36i)9-s + (1.67 + 0.448i)11-s + (0.0999 + 0.758i)17-s + (1.70 + 0.707i)19-s + (0.258 − 0.965i)25-s + (−2.36 − 2.36i)27-s + 3.34i·33-s + (−1.12 − 1.46i)41-s + (−0.448 − 0.258i)43-s + (−0.965 − 0.258i)49-s + (−1.36 + 0.565i)51-s + (−0.465 + 3.53i)57-s + (−0.758 − 0.0999i)59-s + (−1.12 − 0.465i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.634 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.634 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3104\)    =    \(2^{5} \cdot 97\)
Sign: $-0.634 - 0.773i$
Analytic conductor: \(1.54909\)
Root analytic conductor: \(1.24462\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3104} (2255, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3104,\ (\ :0),\ -0.634 - 0.773i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.570348225\)
\(L(\frac12)\) \(\approx\) \(1.570348225\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
97 \( 1 + (0.258 + 0.965i)T \)
good3 \( 1 + (-0.5 - 1.86i)T + (-0.866 + 0.5i)T^{2} \)
5 \( 1 + (-0.258 + 0.965i)T^{2} \)
7 \( 1 + (0.965 + 0.258i)T^{2} \)
11 \( 1 + (-1.67 - 0.448i)T + (0.866 + 0.5i)T^{2} \)
13 \( 1 + (-0.258 + 0.965i)T^{2} \)
17 \( 1 + (-0.0999 - 0.758i)T + (-0.965 + 0.258i)T^{2} \)
19 \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \)
23 \( 1 + (-0.965 + 0.258i)T^{2} \)
29 \( 1 + (0.258 - 0.965i)T^{2} \)
31 \( 1 + (0.866 - 0.5i)T^{2} \)
37 \( 1 + (-0.965 - 0.258i)T^{2} \)
41 \( 1 + (1.12 + 1.46i)T + (-0.258 + 0.965i)T^{2} \)
43 \( 1 + (0.448 + 0.258i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-0.866 + 0.5i)T^{2} \)
59 \( 1 + (0.758 + 0.0999i)T + (0.965 + 0.258i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (1.12 + 0.465i)T + (0.707 + 0.707i)T^{2} \)
71 \( 1 + (0.258 + 0.965i)T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + iT^{2} \)
83 \( 1 + (-0.258 - 1.96i)T + (-0.965 + 0.258i)T^{2} \)
89 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
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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

−9.296470997866164217582457396041, −8.615142044406262418226876743951, −7.969559635710770951515828098882, −6.85361436836872637708646030192, −5.89379660048272708510217606567, −5.14608106372624727472601955117, −4.31359786265935274662880606055, −3.71975299170670309814332859730, −3.10560984405979842812377444113, −1.74121614566766929558505231872, 1.02722502964051098795050805574, 1.62508712363600844694026457862, 3.00520405473588119922837482821, 3.35457065475229030563580767500, 4.88601603173039486235155587288, 5.89768748844911794249348819119, 6.55615746911265343052906834472, 7.15485321638792286962436810345, 7.67168897092172108011427864040, 8.554885493028798666649742014165

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