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Properties

Label 2-1380-345.344-c1-0-39
Degree $2$
Conductor $1380$
Sign $0.356 + 0.934i$
Analytic cond. $11.0193$
Root an. cond. $3.31954$
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  + (1.35 − 1.07i)3-s + (1.70 + 1.44i)5-s − 5.00·7-s + (0.676 − 2.92i)9-s + 5.13·11-s − 5.63i·13-s + (3.87 + 0.121i)15-s − 1.95i·17-s − 3.00i·19-s + (−6.78 + 5.39i)21-s + (1.84 + 4.42i)23-s + (0.819 + 4.93i)25-s + (−2.23 − 4.69i)27-s − 4.77i·29-s − 2.36·31-s + ⋯
L(s)  = 1  + (0.782 − 0.622i)3-s + (0.762 + 0.646i)5-s − 1.89·7-s + (0.225 − 0.974i)9-s + 1.54·11-s − 1.56i·13-s + (0.999 + 0.0313i)15-s − 0.474i·17-s − 0.689i·19-s + (−1.48 + 1.17i)21-s + (0.385 + 0.922i)23-s + (0.163 + 0.986i)25-s + (−0.429 − 0.902i)27-s − 0.886i·29-s − 0.424·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 + 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.356 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.356 + 0.934i$
Analytic conductor: \(11.0193\)
Root analytic conductor: \(3.31954\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :1/2),\ 0.356 + 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.129054491\)
\(L(\frac12)\) \(\approx\) \(2.129054491\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.35 + 1.07i)T \)
5 \( 1 + (-1.70 - 1.44i)T \)
23 \( 1 + (-1.84 - 4.42i)T \)
good7 \( 1 + 5.00T + 7T^{2} \)
11 \( 1 - 5.13T + 11T^{2} \)
13 \( 1 + 5.63iT - 13T^{2} \)
17 \( 1 + 1.95iT - 17T^{2} \)
19 \( 1 + 3.00iT - 19T^{2} \)
29 \( 1 + 4.77iT - 29T^{2} \)
31 \( 1 + 2.36T + 31T^{2} \)
37 \( 1 + 3.30T + 37T^{2} \)
41 \( 1 + 2.97iT - 41T^{2} \)
43 \( 1 - 10.3T + 43T^{2} \)
47 \( 1 - 6.12T + 47T^{2} \)
53 \( 1 + 5.69iT - 53T^{2} \)
59 \( 1 - 9.47iT - 59T^{2} \)
61 \( 1 + 4.89iT - 61T^{2} \)
67 \( 1 - 2.27T + 67T^{2} \)
71 \( 1 + 6.85iT - 71T^{2} \)
73 \( 1 + 14.0iT - 73T^{2} \)
79 \( 1 - 4.83iT - 79T^{2} \)
83 \( 1 + 0.561iT - 83T^{2} \)
89 \( 1 + 8.14T + 89T^{2} \)
97 \( 1 + 4.48T + 97T^{2} \)
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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

−9.327878313792254855410168715028, −8.984489126805878952227457507451, −7.54946568301985080539890404364, −6.96309021287090346887152374012, −6.28748757787942336886529173007, −5.65755405358735911135121403663, −3.77074428478598887611622613621, −3.19096122149900415862988995717, −2.40290266242560812736489912802, −0.827643791367987758044297102396, 1.51161487765241068748322595065, 2.68947845578116723263517327794, 3.85071705446000321791839659547, 4.28051487354573109125448851711, 5.68940359377554795133307335175, 6.50373796321389944719726088011, 7.07338441607164219500177778312, 8.701123124017318956861455498842, 8.994308946650982399890058035523, 9.588830095459308736413848481491

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