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Properties

Label 2-1520-380.379-c1-0-51
Degree $2$
Conductor $1520$
Sign $-0.997 - 0.0754i$
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  − 2.96i·3-s + (−0.602 + 2.15i)5-s + 0.562·7-s − 5.77·9-s − 4.19i·11-s + 0.500·13-s + (6.38 + 1.78i)15-s − 2.59i·17-s + (3.27 − 2.87i)19-s − 1.66i·21-s − 5.09·23-s + (−4.27 − 2.59i)25-s + 8.23i·27-s + 6.71i·29-s + 4.79·31-s + ⋯
L(s)  = 1  − 1.71i·3-s + (−0.269 + 0.963i)5-s + 0.212·7-s − 1.92·9-s − 1.26i·11-s + 0.138·13-s + (1.64 + 0.460i)15-s − 0.628i·17-s + (0.751 − 0.659i)19-s − 0.363i·21-s − 1.06·23-s + (−0.854 − 0.518i)25-s + 1.58i·27-s + 1.24i·29-s + 0.861·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $-0.997 - 0.0754i$
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: \((2,\ 1520,\ (\ :1/2),\ -0.997 - 0.0754i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9186420480\)
\(L(\frac12)\) \(\approx\) \(0.9186420480\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.602 - 2.15i)T \)
19 \( 1 + (-3.27 + 2.87i)T \)
good3 \( 1 + 2.96iT - 3T^{2} \)
7 \( 1 - 0.562T + 7T^{2} \)
11 \( 1 + 4.19iT - 11T^{2} \)
13 \( 1 - 0.500T + 13T^{2} \)
17 \( 1 + 2.59iT - 17T^{2} \)
23 \( 1 + 5.09T + 23T^{2} \)
29 \( 1 - 6.71iT - 29T^{2} \)
31 \( 1 - 4.79T + 31T^{2} \)
37 \( 1 + 3.89T + 37T^{2} \)
41 \( 1 + 11.6iT - 41T^{2} \)
43 \( 1 + 8.46T + 43T^{2} \)
47 \( 1 + 10.2T + 47T^{2} \)
53 \( 1 + 5.73T + 53T^{2} \)
59 \( 1 - 6.83T + 59T^{2} \)
61 \( 1 - 7.25T + 61T^{2} \)
67 \( 1 + 4.43iT - 67T^{2} \)
71 \( 1 + 4.79T + 71T^{2} \)
73 \( 1 - 10.4iT - 73T^{2} \)
79 \( 1 + 7.96T + 79T^{2} \)
83 \( 1 + 2.12T + 83T^{2} \)
89 \( 1 + 8.67iT - 89T^{2} \)
97 \( 1 - 14.1T + 97T^{2} \)
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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

−8.630242214152203936147523250092, −8.178003369228384928836905956136, −7.28969490971118475561801355127, −6.79219514327362599649456597258, −6.07497220706741761759345647751, −5.18529492689017231828830174671, −3.48251428422174322845922307079, −2.81056869513050655098805715160, −1.68028003786391137015713515163, −0.35348661585823347303987468156, 1.73677052887819133750636344361, 3.31083562165751874506799700725, 4.20583853308094293827367807758, 4.68921536510392346471190200779, 5.42099321485503139705249248088, 6.41271073660321919586046449855, 7.972530251979567818478734239015, 8.246006563305389396898672082189, 9.346442214341629999738530838640, 9.942900865586700252433160262888

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