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Properties

Label 2-1520-1520.189-c0-0-7
Degree $2$
Conductor $1520$
Sign $0.923 + 0.382i$
Analytic cond. $0.758578$
Root an. cond. $0.870964$
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  + (0.195 − 0.980i)2-s + (1.38 + 1.38i)3-s + (−0.923 − 0.382i)4-s + (0.707 − 0.707i)5-s + (1.63 − 1.08i)6-s + (−0.555 + 0.831i)8-s + 2.84i·9-s + (−0.555 − 0.831i)10-s + (0.541 − 0.541i)11-s + (−0.750 − 1.81i)12-s + (−1.17 − 1.17i)13-s + 1.96·15-s + (0.707 + 0.707i)16-s + (2.79 + 0.555i)18-s + (0.707 + 0.707i)19-s + (−0.923 + 0.382i)20-s + ⋯
L(s)  = 1  + (0.195 − 0.980i)2-s + (1.38 + 1.38i)3-s + (−0.923 − 0.382i)4-s + (0.707 − 0.707i)5-s + (1.63 − 1.08i)6-s + (−0.555 + 0.831i)8-s + 2.84i·9-s + (−0.555 − 0.831i)10-s + (0.541 − 0.541i)11-s + (−0.750 − 1.81i)12-s + (−1.17 − 1.17i)13-s + 1.96·15-s + (0.707 + 0.707i)16-s + (2.79 + 0.555i)18-s + (0.707 + 0.707i)19-s + (−0.923 + 0.382i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $0.923 + 0.382i$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :0),\ 0.923 + 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.868785328\)
\(L(\frac12)\) \(\approx\) \(1.868785328\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.195 + 0.980i)T \)
5 \( 1 + (-0.707 + 0.707i)T \)
19 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (-1.38 - 1.38i)T + iT^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + (-0.541 + 0.541i)T - iT^{2} \)
13 \( 1 + (1.17 + 1.17i)T + iT^{2} \)
17 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.275 - 0.275i)T - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (1.38 - 1.38i)T - iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
67 \( 1 + (0.785 + 0.785i)T + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - 1.66T + T^{2} \)
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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

−9.437878913920175219118714273158, −9.333122052703425606145644225081, −8.308307233691165577633226324194, −7.81662379637245830524077246488, −5.81604535537368864573379878158, −5.05401335729695709950397498642, −4.46741727891238614802913873201, −3.38716953950274229877931093051, −2.82333960686979936996818769627, −1.69972219851941271192881782158, 1.66953798124886658786403609740, 2.64614815788203547965382413550, 3.56161942812729849777783677692, 4.76157303712136607181817432326, 6.09633997072906205407500912915, 6.84639394924002370322252019475, 7.12669456532535078438428109661, 7.82236053371497753000696388304, 8.855692862300354798482937659867, 9.423141790621730040878596637329

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