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Properties

Label 2-3040-3040.1709-c0-0-6
Degree $2$
Conductor $3040$
Sign $-0.995 + 0.0980i$
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  + (0.0980 − 0.995i)2-s + (0.485 − 1.17i)3-s + (−0.980 − 0.195i)4-s + (0.923 − 0.382i)5-s + (−1.11 − 0.598i)6-s + (−0.290 + 0.956i)8-s + (−0.431 − 0.431i)9-s + (−0.290 − 0.956i)10-s + (−0.636 − 1.53i)11-s + (−0.704 + 1.05i)12-s + (1.62 + 0.674i)13-s − 1.26i·15-s + (0.923 + 0.382i)16-s + (−0.471 + 0.386i)18-s + (−0.923 − 0.382i)19-s + (−0.980 + 0.195i)20-s + ⋯
L(s)  = 1  + (0.0980 − 0.995i)2-s + (0.485 − 1.17i)3-s + (−0.980 − 0.195i)4-s + (0.923 − 0.382i)5-s + (−1.11 − 0.598i)6-s + (−0.290 + 0.956i)8-s + (−0.431 − 0.431i)9-s + (−0.290 − 0.956i)10-s + (−0.636 − 1.53i)11-s + (−0.704 + 1.05i)12-s + (1.62 + 0.674i)13-s − 1.26i·15-s + (0.923 + 0.382i)16-s + (−0.471 + 0.386i)18-s + (−0.923 − 0.382i)19-s + (−0.980 + 0.195i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3040\)    =    \(2^{5} \cdot 5 \cdot 19\)
Sign: $-0.995 + 0.0980i$
Analytic conductor: \(1.51715\)
Root analytic conductor: \(1.23172\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3040} (1709, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3040,\ (\ :0),\ -0.995 + 0.0980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.638883080\)
\(L(\frac12)\) \(\approx\) \(1.638883080\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0980 + 0.995i)T \)
5 \( 1 + (-0.923 + 0.382i)T \)
19 \( 1 + (0.923 + 0.382i)T \)
good3 \( 1 + (-0.485 + 1.17i)T + (-0.707 - 0.707i)T^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + (0.636 + 1.53i)T + (-0.707 + 0.707i)T^{2} \)
13 \( 1 + (-1.62 - 0.674i)T + (0.707 + 0.707i)T^{2} \)
17 \( 1 + T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + (0.707 + 0.707i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (1.83 - 0.761i)T + (0.707 - 0.707i)T^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 + (0.707 - 0.707i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-0.591 - 1.42i)T + (-0.707 + 0.707i)T^{2} \)
59 \( 1 + (-0.707 + 0.707i)T^{2} \)
61 \( 1 + (0.425 - 1.02i)T + (-0.707 - 0.707i)T^{2} \)
67 \( 1 + (0.732 - 1.76i)T + (-0.707 - 0.707i)T^{2} \)
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 \( 1 + 0.942T + T^{2} \)
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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

−8.742537601331100671832347364587, −8.233379943051855607107003182674, −6.97301145483287163186587506923, −6.10577288440349341980078570933, −5.63113177249924713286143902754, −4.51154903351324080162395158526, −3.43925647816447540417854639399, −2.62273278990330296652853729842, −1.74677338652528304268438416038, −1.00961141699652097689882050759, 1.85199848089373443947763673347, 3.18504112448888641683276024999, 3.87271361960550323291041956039, 4.75299912178049922290038530063, 5.39011406517105520144951906072, 6.21992341250932874928130780967, 6.90497701184722944253413711183, 7.83479409911977476370429828418, 8.617668365071651658573204656460, 9.178901576075466242790592578352

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