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Properties

Label 2-1520-380.379-c1-0-44
Degree $2$
Conductor $1520$
Sign $-0.101 + 0.994i$
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  − 2.75i·3-s + (2.03 + 0.928i)5-s + 4.29·7-s − 4.59·9-s − 3.89i·11-s + 1.27·13-s + (2.55 − 5.60i)15-s + 3.77i·17-s + (−4.27 − 0.851i)19-s − 11.8i·21-s − 1.36·23-s + (3.27 + 3.77i)25-s + 4.39i·27-s − 4.02i·29-s + 10.0·31-s + ⋯
L(s)  = 1  − 1.59i·3-s + (0.909 + 0.415i)5-s + 1.62·7-s − 1.53·9-s − 1.17i·11-s + 0.353·13-s + (0.660 − 1.44i)15-s + 0.916i·17-s + (−0.980 − 0.195i)19-s − 2.58i·21-s − 0.284·23-s + (0.654 + 0.755i)25-s + 0.844i·27-s − 0.748i·29-s + 1.80·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.101 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.101 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $-0.101 + 0.994i$
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: \((2,\ 1520,\ (\ :1/2),\ -0.101 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.376566405\)
\(L(\frac12)\) \(\approx\) \(2.376566405\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.03 - 0.928i)T \)
19 \( 1 + (4.27 + 0.851i)T \)
good3 \( 1 + 2.75iT - 3T^{2} \)
7 \( 1 - 4.29T + 7T^{2} \)
11 \( 1 + 3.89iT - 11T^{2} \)
13 \( 1 - 1.27T + 13T^{2} \)
17 \( 1 - 3.77iT - 17T^{2} \)
23 \( 1 + 1.36T + 23T^{2} \)
29 \( 1 + 4.02iT - 29T^{2} \)
31 \( 1 - 10.0T + 31T^{2} \)
37 \( 1 + 8.39T + 37T^{2} \)
41 \( 1 + 6.20iT - 41T^{2} \)
43 \( 1 - 0.671T + 43T^{2} \)
47 \( 1 - 9.54T + 47T^{2} \)
53 \( 1 - 9.95T + 53T^{2} \)
59 \( 1 + 3.08T + 59T^{2} \)
61 \( 1 + 6.74T + 61T^{2} \)
67 \( 1 - 13.4iT - 67T^{2} \)
71 \( 1 + 10.0T + 71T^{2} \)
73 \( 1 - 1.28iT - 73T^{2} \)
79 \( 1 - 4.94T + 79T^{2} \)
83 \( 1 + 11.1T + 83T^{2} \)
89 \( 1 - 0.805iT - 89T^{2} \)
97 \( 1 + 1.15T + 97T^{2} \)
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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

−8.626784025841526378013075890707, −8.566842041691546058153429419560, −7.66621583833764887774603714977, −6.79977612656260134353390520509, −6.02342140051357161212774675484, −5.54330486363476262834435264301, −4.18229470204804581614937982149, −2.65337064366427774711619417506, −1.88596876808209692780398768956, −1.04469315014153154970751093589, 1.55206752781555973603390793824, 2.62239111081443246126207628773, 4.12335981850840435732160748151, 4.78271696426528675372041177401, 5.09950501051641244202322028141, 6.15939230643137366162595737891, 7.38075893218514092046066291153, 8.487238177154335532246362390414, 8.903336127311570419512878351384, 9.792235909772801655509061126664

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