Properties

Label 2-1520-380.379-c1-0-1
Degree $2$
Conductor $1520$
Sign $0.363 + 0.931i$
Analytic cond. $12.1372$
Root an. cond. $3.48385$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.59i·3-s + (−1.73 + 1.41i)5-s − 3.38·7-s − 3.73·9-s + 1.75i·11-s − 6.21·13-s + (−3.66 − 4.49i)15-s + 6.69i·17-s + (1.34 − 4.14i)19-s − 8.78i·21-s + 5.86·23-s + (0.999 − 4.89i)25-s − 1.89i·27-s − 6.43i·29-s + 6.35·31-s + ⋯
L(s)  = 1  + 1.49i·3-s + (−0.774 + 0.632i)5-s − 1.27·7-s − 1.24·9-s + 0.528i·11-s − 1.72·13-s + (−0.947 − 1.16i)15-s + 1.62i·17-s + (0.308 − 0.951i)19-s − 1.91i·21-s + 1.22·23-s + (0.199 − 0.979i)25-s − 0.365i·27-s − 1.19i·29-s + 1.14·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.363 + 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.363 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $0.363 + 0.931i$
Analytic conductor: \(12.1372\)
Root analytic conductor: \(3.48385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (1519, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :1/2),\ 0.363 + 0.931i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1207974913\)
\(L(\frac12)\) \(\approx\) \(0.1207974913\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (1.73 - 1.41i)T \)
19 \( 1 + (-1.34 + 4.14i)T \)
good3 \( 1 - 2.59iT - 3T^{2} \)
7 \( 1 + 3.38T + 7T^{2} \)
11 \( 1 - 1.75iT - 11T^{2} \)
13 \( 1 + 6.21T + 13T^{2} \)
17 \( 1 - 6.69iT - 17T^{2} \)
23 \( 1 - 5.86T + 23T^{2} \)
29 \( 1 + 6.43iT - 29T^{2} \)
31 \( 1 - 6.35T + 31T^{2} \)
37 \( 1 + 1.66T + 37T^{2} \)
41 \( 1 - 8.78iT - 41T^{2} \)
43 \( 1 + 0.907T + 43T^{2} \)
47 \( 1 - 10.1T + 47T^{2} \)
53 \( 1 + 10.7T + 53T^{2} \)
59 \( 1 + 6.35T + 59T^{2} \)
61 \( 1 + 2.19T + 61T^{2} \)
67 \( 1 + 4.49iT - 67T^{2} \)
71 \( 1 - 1.70T + 71T^{2} \)
73 \( 1 + 3.58iT - 73T^{2} \)
79 \( 1 + 16.3T + 79T^{2} \)
83 \( 1 + 1.57T + 83T^{2} \)
89 \( 1 - 6.43iT - 89T^{2} \)
97 \( 1 - 1.66T + 97T^{2} \)
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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

−9.882281636533500323723802594619, −9.716356114166178853745498002341, −8.742055120268927681150848452471, −7.67442918321786382041428069260, −6.88218648253434705228069105003, −6.05998517379805158299057800109, −4.75959817914828075725552278252, −4.33020922673454500342416247906, −3.25898126685532122428553125240, −2.70303441704271760714896464355, 0.05592804851430324940125594257, 1.03108548258976671259165534176, 2.61136501508570325323977780119, 3.29638163669517986961824897984, 4.75908876858009940962148278866, 5.58922113625674467170401769671, 6.70721262235153613948770085287, 7.25706099338011208607214940454, 7.69993965935105895405060923118, 8.802989267143876088366908484060

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