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Properties

Label 2-390-65.4-c1-0-10
Degree $2$
Conductor $390$
Sign $-0.956 + 0.290i$
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 + (−0.866 − 0.5i)3-s + (−0.499 − 0.866i)4-s + (0.230 − 2.22i)5-s + (−0.866 + 0.499i)6-s + (−0.432 − 0.749i)7-s − 0.999·8-s + (0.499 + 0.866i)9-s + (−1.81 − 1.31i)10-s + (0.151 + 0.0874i)11-s + 0.999i·12-s + (−1.35 − 3.34i)13-s − 0.865·14-s + (−1.31 + 1.81i)15-s + (−0.5 + 0.866i)16-s + (−7.08 + 4.08i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.499 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (0.103 − 0.994i)5-s + (−0.353 + 0.204i)6-s + (−0.163 − 0.283i)7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (−0.572 − 0.414i)10-s + (0.0456 + 0.0263i)11-s + 0.288i·12-s + (−0.375 − 0.926i)13-s − 0.231·14-s + (−0.338 + 0.467i)15-s + (−0.125 + 0.216i)16-s + (−1.71 + 0.991i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 + 0.290i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.956 + 0.290i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ -0.956 + 0.290i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.158118 - 1.06479i\)
\(L(\frac12)\) \(\approx\) \(0.158118 - 1.06479i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (-0.230 + 2.22i)T \)
13 \( 1 + (1.35 + 3.34i)T \)
good7 \( 1 + (0.432 + 0.749i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.151 - 0.0874i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (7.08 - 4.08i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.20 + 3.00i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.52 + 1.45i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.24 + 5.62i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.95iT - 31T^{2} \)
37 \( 1 + (-0.879 + 1.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (7.08 + 4.08i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-7.94 + 4.58i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 11.9T + 47T^{2} \)
53 \( 1 + 2.48iT - 53T^{2} \)
59 \( 1 + (6.09 - 3.51i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.98 - 6.90i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.36 - 2.36i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-12.2 + 7.08i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 12.8T + 73T^{2} \)
79 \( 1 - 9.48T + 79T^{2} \)
83 \( 1 + 0.139T + 83T^{2} \)
89 \( 1 + (11.3 + 6.56i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.32 - 7.48i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.92526056635595335630364033791, −10.22899612486075784587306078043, −9.159725648501165945248947198059, −8.266655011758995178735316298537, −6.99866101719093589300629729598, −5.83004697902622475560416325551, −4.95429994461003166821891841269, −3.97119591157888938428026562264, −2.26320642899661455887991358574, −0.67008528692303149694422390417, 2.53267067372415735888785885212, 3.90229530435769496877178585078, 5.00522997830808365149211542829, 6.13692377414886330989178152263, 6.82808634598637316244050913804, 7.68644700911757678406309813848, 9.159763003996106786383828602506, 9.765040244887583397792532232413, 11.03077797900610063956136943144, 11.62481248304269502695483946761

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