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Properties

Label 2-390-15.2-c1-0-12
Degree $2$
Conductor $390$
Sign $0.749 + 0.662i$
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  + (0.707 − 0.707i)2-s + (1.41 − i)3-s − 1.00i·4-s + (−0.707 + 2.12i)5-s + (0.292 − 1.70i)6-s + (3.41 + 3.41i)7-s + (−0.707 − 0.707i)8-s + (1.00 − 2.82i)9-s + (0.999 + 2i)10-s − 4i·11-s + (−1.00 − 1.41i)12-s + (0.707 − 0.707i)13-s + 4.82·14-s + (1.12 + 3.70i)15-s − 1.00·16-s + (−3 + 3i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.816 − 0.577i)3-s − 0.500i·4-s + (−0.316 + 0.948i)5-s + (0.119 − 0.696i)6-s + (1.29 + 1.29i)7-s + (−0.250 − 0.250i)8-s + (0.333 − 0.942i)9-s + (0.316 + 0.632i)10-s − 1.20i·11-s + (−0.288 − 0.408i)12-s + (0.196 − 0.196i)13-s + 1.29·14-s + (0.289 + 0.957i)15-s − 0.250·16-s + (−0.727 + 0.727i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.749 + 0.662i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.749 + 0.662i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.19311 - 0.829868i\)
\(L(\frac12)\) \(\approx\) \(2.19311 - 0.829868i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (-1.41 + i)T \)
5 \( 1 + (0.707 - 2.12i)T \)
13 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 + (-3.41 - 3.41i)T + 7iT^{2} \)
11 \( 1 + 4iT - 11T^{2} \)
17 \( 1 + (3 - 3i)T - 17iT^{2} \)
19 \( 1 + 1.17iT - 19T^{2} \)
23 \( 1 + (-2.82 - 2.82i)T + 23iT^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 0.585T + 31T^{2} \)
37 \( 1 + (5.41 + 5.41i)T + 37iT^{2} \)
41 \( 1 + 2iT - 41T^{2} \)
43 \( 1 + (7.24 - 7.24i)T - 43iT^{2} \)
47 \( 1 + (-6.24 + 6.24i)T - 47iT^{2} \)
53 \( 1 + (-3.41 - 3.41i)T + 53iT^{2} \)
59 \( 1 + 12.4T + 59T^{2} \)
61 \( 1 + 0.828T + 61T^{2} \)
67 \( 1 + (-4.82 - 4.82i)T + 67iT^{2} \)
71 \( 1 - 5.75iT - 71T^{2} \)
73 \( 1 + (-3.07 + 3.07i)T - 73iT^{2} \)
79 \( 1 - 4.48iT - 79T^{2} \)
83 \( 1 + (1.17 + 1.17i)T + 83iT^{2} \)
89 \( 1 - 16.8T + 89T^{2} \)
97 \( 1 + (5.07 + 5.07i)T + 97iT^{2} \)
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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

−11.33107079060415504418538585469, −10.68999849366436104053427939763, −9.136049092186417301058590747544, −8.503112460212929820745317400306, −7.59411723170592654912278126469, −6.35229711959332705653023720912, −5.42836640169361616246243333882, −3.84458850088418531275537707719, −2.82863775180042371616231667831, −1.82401360944476889042286792134, 1.81780717236730009860813051572, 3.76758878223743538022469706662, 4.64478318045562385908415818609, 4.96910290928954594616479780744, 7.05364071541796512982191358165, 7.68985478775935287237194826593, 8.507049963732655248336915482686, 9.389391070794083706329646729479, 10.51110583168260453981805279624, 11.41627581750830790389237105383

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