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Properties

Label 2-390-65.58-c1-0-13
Degree $2$
Conductor $390$
Sign $-0.108 + 0.994i$
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  + (−0.866 − 0.5i)2-s + (0.965 − 0.258i)3-s + (0.499 + 0.866i)4-s + (1.64 − 1.50i)5-s + (−0.965 − 0.258i)6-s + (−1.56 − 2.70i)7-s − 0.999i·8-s + (0.866 − 0.499i)9-s + (−2.18 + 0.482i)10-s + (0.628 − 0.168i)11-s + (0.707 + 0.707i)12-s + (−3.13 − 1.78i)13-s + 3.12i·14-s + (1.20 − 1.88i)15-s + (−0.5 + 0.866i)16-s + (−0.0226 + 0.0844i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.557 − 0.149i)3-s + (0.249 + 0.433i)4-s + (0.737 − 0.674i)5-s + (−0.394 − 0.105i)6-s + (−0.589 − 1.02i)7-s − 0.353i·8-s + (0.288 − 0.166i)9-s + (−0.690 + 0.152i)10-s + (0.189 − 0.0507i)11-s + (0.204 + 0.204i)12-s + (−0.868 − 0.495i)13-s + 0.834i·14-s + (0.310 − 0.486i)15-s + (−0.125 + 0.216i)16-s + (−0.00548 + 0.0204i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.108 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.108 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.108 + 0.994i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ -0.108 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.823721 - 0.918109i\)
\(L(\frac12)\) \(\approx\) \(0.823721 - 0.918109i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 + (-1.64 + 1.50i)T \)
13 \( 1 + (3.13 + 1.78i)T \)
good7 \( 1 + (1.56 + 2.70i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.628 + 0.168i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (0.0226 - 0.0844i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.264 + 0.986i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.611 - 2.28i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-7.96 - 4.59i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.40 - 1.40i)T - 31iT^{2} \)
37 \( 1 + (-3.80 + 6.58i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.901 + 3.36i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (9.73 + 2.60i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 - 4.74T + 47T^{2} \)
53 \( 1 + (-6.10 - 6.10i)T + 53iT^{2} \)
59 \( 1 + (12.7 + 3.42i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.40 - 5.89i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.39 - 5.42i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-13.0 - 3.48i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + 2.45iT - 73T^{2} \)
79 \( 1 - 10.4iT - 79T^{2} \)
83 \( 1 - 1.51T + 83T^{2} \)
89 \( 1 + (-4.84 - 18.0i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-4.43 + 2.55i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.71036990976981896817827824817, −10.03197427611681844145907332642, −9.344794532942762770992532070341, −8.486528631575719018215697593909, −7.43083071355733957813287781319, −6.61477567749658813170118142691, −5.13192453664950742565873184890, −3.78562168520669425983581687602, −2.50184291955474764851707256140, −0.969366924413690423518343123371, 2.11159802674323062282264667645, 3.02205165488721833084930689013, 4.86255608339410565344452880213, 6.15532433494367660226896008412, 6.75853327067023159897647382951, 7.969958690335186330444253804484, 8.947581812776139326364711342316, 9.719944904925127036462147976651, 10.16455774493394015763813594623, 11.45967626246842961358054721038

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