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Properties

Label 2-1520-5.4-c1-0-28
Degree $2$
Conductor $1520$
Sign $0.981 + 0.193i$
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  + 2.76i·3-s + (−2.19 − 0.432i)5-s + 0.761i·7-s − 4.62·9-s + 0.864·11-s − 5.62i·13-s + (1.19 − 6.05i)15-s − 3.62i·17-s + 19-s − 2.10·21-s − 8.01i·23-s + (4.62 + 1.89i)25-s − 4.49i·27-s + 7.35·29-s − 8.11·31-s + ⋯
L(s)  = 1  + 1.59i·3-s + (−0.981 − 0.193i)5-s + 0.287i·7-s − 1.54·9-s + 0.260·11-s − 1.56i·13-s + (0.308 − 1.56i)15-s − 0.879i·17-s + 0.229·19-s − 0.458·21-s − 1.67i·23-s + (0.925 + 0.379i)25-s − 0.864i·27-s + 1.36·29-s − 1.45·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.193i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 + 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $0.981 + 0.193i$
Analytic conductor: \(12.1372\)
Root analytic conductor: \(3.48385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (609, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :1/2),\ 0.981 + 0.193i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.024812227\)
\(L(\frac12)\) \(\approx\) \(1.024812227\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.19 + 0.432i)T \)
19 \( 1 - T \)
good3 \( 1 - 2.76iT - 3T^{2} \)
7 \( 1 - 0.761iT - 7T^{2} \)
11 \( 1 - 0.864T + 11T^{2} \)
13 \( 1 + 5.62iT - 13T^{2} \)
17 \( 1 + 3.62iT - 17T^{2} \)
23 \( 1 + 8.01iT - 23T^{2} \)
29 \( 1 - 7.35T + 29T^{2} \)
31 \( 1 + 8.11T + 31T^{2} \)
37 \( 1 - 0.476iT - 37T^{2} \)
41 \( 1 + 2.65T + 41T^{2} \)
43 \( 1 - 6.86iT - 43T^{2} \)
47 \( 1 + 1.25iT - 47T^{2} \)
53 \( 1 + 2.37iT - 53T^{2} \)
59 \( 1 - 4.49T + 59T^{2} \)
61 \( 1 + 10.8T + 61T^{2} \)
67 \( 1 + 1.03iT - 67T^{2} \)
71 \( 1 - 10.1T + 71T^{2} \)
73 \( 1 + 16.4iT - 73T^{2} \)
79 \( 1 + 12.5T + 79T^{2} \)
83 \( 1 + 0.270iT - 83T^{2} \)
89 \( 1 + 0.387T + 89T^{2} \)
97 \( 1 + 8.50iT - 97T^{2} \)
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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

−9.440707218909632647153523044838, −8.711790625431252400764137168390, −8.126085442178349881013645709402, −7.13349575892993182826001426775, −5.90600592816345173014002974984, −4.96696639998972070929361279695, −4.51330608312517215419253441700, −3.44721789293473496451225606717, −2.85247742512199678575014759264, −0.46548561277502465146308099783, 1.16108099479652404501990489453, 2.09074355350196077726288605815, 3.45308900814729654949729539519, 4.26658726901594697689314689922, 5.60881130709444336705244582374, 6.61707641288375917417945335691, 7.09994688384913060962331227721, 7.67335336638814681001159006568, 8.505437073438345229443507678414, 9.198866205076039835458387462695

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