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Properties

Label 2-3104-776.419-c0-0-0
Degree $2$
Conductor $3104$
Sign $-0.997 + 0.0717i$
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  + (0.207 − 1.57i)3-s + (−1.46 − 0.392i)9-s + (−1.53 − 1.17i)11-s + (0.172 − 0.349i)17-s + (0.923 − 0.617i)19-s + (−0.991 + 0.130i)25-s + (−0.314 + 0.758i)27-s + (−2.16 + 2.16i)33-s + (−1.09 + 1.25i)41-s + (−0.252 + 0.0675i)43-s + (0.608 − 0.793i)49-s + (−0.514 − 0.343i)51-s + (−0.779 − 1.58i)57-s + (−1.85 − 0.630i)59-s + (0.732 + 1.09i)67-s + ⋯
L(s)  = 1  + (0.207 − 1.57i)3-s + (−1.46 − 0.392i)9-s + (−1.53 − 1.17i)11-s + (0.172 − 0.349i)17-s + (0.923 − 0.617i)19-s + (−0.991 + 0.130i)25-s + (−0.314 + 0.758i)27-s + (−2.16 + 2.16i)33-s + (−1.09 + 1.25i)41-s + (−0.252 + 0.0675i)43-s + (0.608 − 0.793i)49-s + (−0.514 − 0.343i)51-s + (−0.779 − 1.58i)57-s + (−1.85 − 0.630i)59-s + (0.732 + 1.09i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0717i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3104\)    =    \(2^{5} \cdot 97\)
Sign: $-0.997 + 0.0717i$
Analytic conductor: \(1.54909\)
Root analytic conductor: \(1.24462\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3104} (1583, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3104,\ (\ :0),\ -0.997 + 0.0717i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9169285513\)
\(L(\frac12)\) \(\approx\) \(0.9169285513\)
\(L(1)\) 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 \( 1 + (-0.130 + 0.991i)T \)
good3 \( 1 + (-0.207 + 1.57i)T + (-0.965 - 0.258i)T^{2} \)
5 \( 1 + (0.991 - 0.130i)T^{2} \)
7 \( 1 + (-0.608 + 0.793i)T^{2} \)
11 \( 1 + (1.53 + 1.17i)T + (0.258 + 0.965i)T^{2} \)
13 \( 1 + (-0.991 + 0.130i)T^{2} \)
17 \( 1 + (-0.172 + 0.349i)T + (-0.608 - 0.793i)T^{2} \)
19 \( 1 + (-0.923 + 0.617i)T + (0.382 - 0.923i)T^{2} \)
23 \( 1 + (-0.793 + 0.608i)T^{2} \)
29 \( 1 + (-0.130 - 0.991i)T^{2} \)
31 \( 1 + (-0.965 - 0.258i)T^{2} \)
37 \( 1 + (0.793 + 0.608i)T^{2} \)
41 \( 1 + (1.09 - 1.25i)T + (-0.130 - 0.991i)T^{2} \)
43 \( 1 + (0.252 - 0.0675i)T + (0.866 - 0.5i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (-0.258 + 0.965i)T^{2} \)
59 \( 1 + (1.85 + 0.630i)T + (0.793 + 0.608i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.732 - 1.09i)T + (-0.382 + 0.923i)T^{2} \)
71 \( 1 + (0.130 - 0.991i)T^{2} \)
73 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
79 \( 1 + (0.707 - 0.707i)T^{2} \)
83 \( 1 + (0.576 + 0.284i)T + (0.608 + 0.793i)T^{2} \)
89 \( 1 + (-1.83 + 0.758i)T + (0.707 - 0.707i)T^{2} \)
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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

−8.201171545081573705160975744551, −7.79200902214044221039846445292, −7.17925187341987224003327308683, −6.30669265852939503182811817787, −5.63981515693896875643533856751, −4.89460561593206780591663291239, −3.31079939952567523199389457949, −2.77031535478850142283710161826, −1.74475606756843662744994391208, −0.51618542830514667659849222074, 1.99713116621883188264679205476, 3.01788319148012743773732324003, 3.81395619616716789500916914965, 4.62198138137240264340113371652, 5.25140682587782597646847139384, 5.87060502183431735992020200982, 7.21213442690608933470124567239, 7.84811585891493344156259040458, 8.574053671124547759848465132032, 9.552830848650944757394277582095

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