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Properties

Label 2-390-65.37-c1-0-3
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: $\chi_{390} (37, \cdot )$
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

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

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