Properties

Label 2-3040-3040.189-c0-0-7
Degree $2$
Conductor $3040$
Sign $-0.956 - 0.290i$
Analytic cond. $1.51715$
Root an. cond. $1.23172$
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  + (0.471 − 0.881i)2-s + (−0.536 − 0.222i)3-s + (−0.555 − 0.831i)4-s + (−0.382 − 0.923i)5-s + (−0.448 + 0.368i)6-s + (−0.995 + 0.0980i)8-s + (−0.468 − 0.468i)9-s + (−0.995 − 0.0980i)10-s + (1.81 − 0.750i)11-s + (0.113 + 0.569i)12-s + (0.591 − 1.42i)13-s + 0.580i·15-s + (−0.382 + 0.923i)16-s + (−0.634 + 0.192i)18-s + (0.382 − 0.923i)19-s + (−0.555 + 0.831i)20-s + ⋯
L(s)  = 1  + (0.471 − 0.881i)2-s + (−0.536 − 0.222i)3-s + (−0.555 − 0.831i)4-s + (−0.382 − 0.923i)5-s + (−0.448 + 0.368i)6-s + (−0.995 + 0.0980i)8-s + (−0.468 − 0.468i)9-s + (−0.995 − 0.0980i)10-s + (1.81 − 0.750i)11-s + (0.113 + 0.569i)12-s + (0.591 − 1.42i)13-s + 0.580i·15-s + (−0.382 + 0.923i)16-s + (−0.634 + 0.192i)18-s + (0.382 − 0.923i)19-s + (−0.555 + 0.831i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3040\)    =    \(2^{5} \cdot 5 \cdot 19\)
Sign: $-0.956 - 0.290i$
Analytic conductor: \(1.51715\)
Root analytic conductor: \(1.23172\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3040} (189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3040,\ (\ :0),\ -0.956 - 0.290i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.131511972\)
\(L(\frac12)\) \(\approx\) \(1.131511972\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.471 + 0.881i)T \)
5 \( 1 + (0.382 + 0.923i)T \)
19 \( 1 + (-0.382 + 0.923i)T \)
good3 \( 1 + (0.536 + 0.222i)T + (0.707 + 0.707i)T^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + (-1.81 + 0.750i)T + (0.707 - 0.707i)T^{2} \)
13 \( 1 + (-0.591 + 1.42i)T + (-0.707 - 0.707i)T^{2} \)
17 \( 1 + T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + (-0.707 - 0.707i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-0.674 - 1.62i)T + (-0.707 + 0.707i)T^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 + (-0.707 + 0.707i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (1.76 - 0.732i)T + (0.707 - 0.707i)T^{2} \)
59 \( 1 + (0.707 - 0.707i)T^{2} \)
61 \( 1 + (0.360 + 0.149i)T + (0.707 + 0.707i)T^{2} \)
67 \( 1 + (0.181 + 0.0750i)T + (0.707 + 0.707i)T^{2} \)
71 \( 1 + iT^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (0.707 + 0.707i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 + 1.26T + T^{2} \)
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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

−8.766450460323700511248448641118, −7.976238573384390096896895022004, −6.59429381753107160082584367005, −6.07138773136718411965360601303, −5.33877021886169900103624505017, −4.56225085477445754843379378836, −3.59033774176315105235661203267, −3.07588300972470452744340532686, −1.32532505189147106411326753185, −0.74025334897137071275010847702, 1.88576112041199998146531585560, 3.26115164921092015052701762967, 4.13029898749270115987058702410, 4.48198817898417787544505424680, 5.79469626047336333373769777045, 6.27218789650653308565894848158, 6.90705156477841191964613050333, 7.55220497763297888215987764063, 8.424042360580877809345696405946, 9.266247214721606424274278011987

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