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Properties

Label 2-390-65.32-c1-0-6
Degree $2$
Conductor $390$
Sign $0.794 - 0.606i$
Analytic cond. $3.11416$
Root an. cond. $1.76469$
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  + (−0.5 + 0.866i)2-s + (0.258 + 0.965i)3-s + (−0.499 − 0.866i)4-s + (1.13 − 1.92i)5-s + (−0.965 − 0.258i)6-s + (−0.245 + 0.141i)7-s + 0.999·8-s + (−0.866 + 0.499i)9-s + (1.10 + 1.94i)10-s + (1.95 − 0.524i)11-s + (0.707 − 0.707i)12-s + (3.31 − 1.42i)13-s − 0.283i·14-s + (2.15 + 0.593i)15-s + (−0.5 + 0.866i)16-s + (2.84 + 0.761i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.149 + 0.557i)3-s + (−0.249 − 0.433i)4-s + (0.505 − 0.862i)5-s + (−0.394 − 0.105i)6-s + (−0.0927 + 0.0535i)7-s + 0.353·8-s + (−0.288 + 0.166i)9-s + (0.349 + 0.614i)10-s + (0.590 − 0.158i)11-s + (0.204 − 0.204i)12-s + (0.918 − 0.394i)13-s − 0.0757i·14-s + (0.556 + 0.153i)15-s + (−0.125 + 0.216i)16-s + (0.689 + 0.184i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 - 0.606i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.794 - 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.794 - 0.606i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.794 - 0.606i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26818 + 0.428676i\)
\(L(\frac12)\) \(\approx\) \(1.26818 + 0.428676i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (-0.258 - 0.965i)T \)
5 \( 1 + (-1.13 + 1.92i)T \)
13 \( 1 + (-3.31 + 1.42i)T \)
good7 \( 1 + (0.245 - 0.141i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.95 + 0.524i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-2.84 - 0.761i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.733 - 2.73i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-8.32 + 2.23i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-0.292 - 0.168i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.64 - 4.64i)T - 31iT^{2} \)
37 \( 1 + (-0.230 - 0.133i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.25 + 4.69i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (2.95 - 11.0i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 - 1.09iT - 47T^{2} \)
53 \( 1 + (-4.98 + 4.98i)T - 53iT^{2} \)
59 \( 1 + (-2.10 - 0.564i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (5.46 + 9.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.86 + 11.8i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (9.81 + 2.63i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + 5.93T + 73T^{2} \)
79 \( 1 - 5.53iT - 79T^{2} \)
83 \( 1 + 6.33iT - 83T^{2} \)
89 \( 1 + (-1.14 - 4.28i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (9.11 + 15.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.14886399757929457085068460574, −10.30010190363371909896405676468, −9.340128451476950189739182259021, −8.768233449552111216521364635009, −7.953651202058900372555704766095, −6.53785533804218667156558797128, −5.64272146107813886083872553902, −4.73692841320649005333488345835, −3.41541239569160661641220453078, −1.31022295806405919891330553172, 1.41742244576682025331762993897, 2.75588368472034051670457658908, 3.80552788442232155575162057741, 5.53335393579069066538022326964, 6.73821810297714412795865585992, 7.34628587558345571360961170870, 8.700629875891997858332318364290, 9.378390301856300869063591154932, 10.39174548602438840096106458628, 11.26152197050286441974891244598

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