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Properties

Label 2-1360-20.3-c1-0-22
Degree $2$
Conductor $1360$
Sign $0.365 - 0.930i$
Analytic cond. $10.8596$
Root an. cond. $3.29539$
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  + (−0.656 + 0.656i)3-s + (1.83 + 1.27i)5-s + (0.769 + 0.769i)7-s + 2.13i·9-s − 3.97i·11-s + (1.99 + 1.99i)13-s + (−2.04 + 0.370i)15-s + (0.707 − 0.707i)17-s + 4.72·19-s − 1.00·21-s + (5.54 − 5.54i)23-s + (1.75 + 4.68i)25-s + (−3.37 − 3.37i)27-s + 1.45i·29-s + 10.2i·31-s + ⋯
L(s)  = 1  + (−0.378 + 0.378i)3-s + (0.822 + 0.569i)5-s + (0.290 + 0.290i)7-s + 0.712i·9-s − 1.19i·11-s + (0.553 + 0.553i)13-s + (−0.527 + 0.0957i)15-s + (0.171 − 0.171i)17-s + 1.08·19-s − 0.220·21-s + (1.15 − 1.15i)23-s + (0.351 + 0.936i)25-s + (−0.649 − 0.649i)27-s + 0.269i·29-s + 1.84i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.365 - 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.365 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1360\)    =    \(2^{4} \cdot 5 \cdot 17\)
Sign: $0.365 - 0.930i$
Analytic conductor: \(10.8596\)
Root analytic conductor: \(3.29539\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1360} (783, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1360,\ (\ :1/2),\ 0.365 - 0.930i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.847072344\)
\(L(\frac12)\) \(\approx\) \(1.847072344\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.83 - 1.27i)T \)
17 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (0.656 - 0.656i)T - 3iT^{2} \)
7 \( 1 + (-0.769 - 0.769i)T + 7iT^{2} \)
11 \( 1 + 3.97iT - 11T^{2} \)
13 \( 1 + (-1.99 - 1.99i)T + 13iT^{2} \)
19 \( 1 - 4.72T + 19T^{2} \)
23 \( 1 + (-5.54 + 5.54i)T - 23iT^{2} \)
29 \( 1 - 1.45iT - 29T^{2} \)
31 \( 1 - 10.2iT - 31T^{2} \)
37 \( 1 + (5.72 - 5.72i)T - 37iT^{2} \)
41 \( 1 + 0.799T + 41T^{2} \)
43 \( 1 + (0.242 - 0.242i)T - 43iT^{2} \)
47 \( 1 + (8.04 + 8.04i)T + 47iT^{2} \)
53 \( 1 + (-1.37 - 1.37i)T + 53iT^{2} \)
59 \( 1 - 3.92T + 59T^{2} \)
61 \( 1 - 13.8T + 61T^{2} \)
67 \( 1 + (-1.73 - 1.73i)T + 67iT^{2} \)
71 \( 1 - 9.93iT - 71T^{2} \)
73 \( 1 + (2.58 + 2.58i)T + 73iT^{2} \)
79 \( 1 + 12.5T + 79T^{2} \)
83 \( 1 + (5.64 - 5.64i)T - 83iT^{2} \)
89 \( 1 - 6.69iT - 89T^{2} \)
97 \( 1 + (7.60 - 7.60i)T - 97iT^{2} \)
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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

−10.00698042975901317667339535403, −8.792589798240063141925851379019, −8.452171480947854014812177805912, −7.06814153392953475294998970859, −6.48189363611543533345790566162, −5.30050013586076153767827414146, −5.14335024371127220456165571432, −3.55898165761405757126235669354, −2.68073905965793354090387479759, −1.36466260393689145476235200336, 0.921616158972701692642297781229, 1.83825771098622636459728167615, 3.30116181622953278391109499753, 4.45372628081157303879501824153, 5.43397176448348663482263730170, 5.98130786557161717842257541443, 7.06310399665976189557972709279, 7.64050248735431362296145449706, 8.759069731115166223349941164315, 9.615170182383626471767778138592

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