Properties

Label 2-3040-3040.189-c0-0-1
Degree $2$
Conductor $3040$
Sign $-0.290 + 0.956i$
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  + (−0.881 − 0.471i)2-s + (−1.76 − 0.732i)3-s + (0.555 + 0.831i)4-s + (−0.382 − 0.923i)5-s + (1.21 + 1.47i)6-s + (−0.0980 − 0.995i)8-s + (1.88 + 1.88i)9-s + (−0.0980 + 0.995i)10-s + (−1.81 + 0.750i)11-s + (−0.373 − 1.87i)12-s + (−0.485 + 1.17i)13-s + 1.91i·15-s + (−0.382 + 0.923i)16-s + (−0.773 − 2.54i)18-s + (0.382 − 0.923i)19-s + (0.555 − 0.831i)20-s + ⋯
L(s)  = 1  + (−0.881 − 0.471i)2-s + (−1.76 − 0.732i)3-s + (0.555 + 0.831i)4-s + (−0.382 − 0.923i)5-s + (1.21 + 1.47i)6-s + (−0.0980 − 0.995i)8-s + (1.88 + 1.88i)9-s + (−0.0980 + 0.995i)10-s + (−1.81 + 0.750i)11-s + (−0.373 − 1.87i)12-s + (−0.485 + 1.17i)13-s + 1.91i·15-s + (−0.382 + 0.923i)16-s + (−0.773 − 2.54i)18-s + (0.382 − 0.923i)19-s + (0.555 − 0.831i)20-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3040\)    =    \(2^{5} \cdot 5 \cdot 19\)
Sign: $-0.290 + 0.956i$
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: \((2,\ 3040,\ (\ :0),\ -0.290 + 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2391855802\)
\(L(\frac12)\) \(\approx\) \(0.2391855802\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.881 + 0.471i)T \)
5 \( 1 + (0.382 + 0.923i)T \)
19 \( 1 + (-0.382 + 0.923i)T \)
good3 \( 1 + (1.76 + 0.732i)T + (0.707 + 0.707i)T^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + (1.81 - 0.750i)T + (0.707 - 0.707i)T^{2} \)
13 \( 1 + (0.485 - 1.17i)T + (-0.707 - 0.707i)T^{2} \)
17 \( 1 + T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + (-0.707 - 0.707i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-0.360 - 0.871i)T + (-0.707 + 0.707i)T^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 + (-0.707 + 0.707i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-0.536 + 0.222i)T + (0.707 - 0.707i)T^{2} \)
59 \( 1 + (0.707 - 0.707i)T^{2} \)
61 \( 1 + (-0.360 - 0.149i)T + (0.707 + 0.707i)T^{2} \)
67 \( 1 + (-1.83 - 0.761i)T + (0.707 + 0.707i)T^{2} \)
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 \( 1 + 1.54T + T^{2} \)
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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

−8.561229181230529936924675379037, −7.81924061190459013771573638871, −7.18399810205391557498184554873, −6.73705718142071797804944728145, −5.52735816046710826123465209247, −4.92244769193089299805184142305, −4.26110914701885710240146591955, −2.48288826424062129031633511474, −1.62671737533164772316018230572, −0.42672120605197690185392452213, 0.67374929746799903186861107987, 2.57829329223365177149594870343, 3.63358228575017389180424002250, 4.94006543476698384797982166720, 5.55494785245930491708754409672, 5.95262681316086185749032754465, 6.82188534972725762051161820830, 7.68302200491340842892517720577, 8.039324249856500161401848888755, 9.408327073869981280016946853176

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