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Properties

Label 2-390-65.58-c1-0-4
Degree $2$
Conductor $390$
Sign $0.976 - 0.213i$
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 + (1.48 + 1.66i)5-s + (−0.965 − 0.258i)6-s + (0.328 + 0.568i)7-s − 0.999i·8-s + (0.866 − 0.499i)9-s + (−0.455 − 2.18i)10-s + (−1.26 + 0.337i)11-s + (0.707 + 0.707i)12-s + (3.54 + 0.656i)13-s − 0.656i·14-s + (1.87 + 1.22i)15-s + (−0.5 + 0.866i)16-s + (−1.47 + 5.49i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.557 − 0.149i)3-s + (0.249 + 0.433i)4-s + (0.665 + 0.746i)5-s + (−0.394 − 0.105i)6-s + (0.124 + 0.214i)7-s − 0.353i·8-s + (0.288 − 0.166i)9-s + (−0.144 − 0.692i)10-s + (−0.379 + 0.101i)11-s + (0.204 + 0.204i)12-s + (0.983 + 0.181i)13-s − 0.175i·14-s + (0.482 + 0.316i)15-s + (−0.125 + 0.216i)16-s + (−0.357 + 1.33i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.213i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 - 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.976 - 0.213i$
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: \((2,\ 390,\ (\ :1/2),\ 0.976 - 0.213i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38328 + 0.149532i\)
\(L(\frac12)\) \(\approx\) \(1.38328 + 0.149532i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.48 - 1.66i)T \)
13 \( 1 + (-3.54 - 0.656i)T \)
good7 \( 1 + (-0.328 - 0.568i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.26 - 0.337i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.47 - 5.49i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.425 + 1.58i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.0557 + 0.208i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-0.0656 - 0.0378i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.13 + 5.13i)T - 31iT^{2} \)
37 \( 1 + (-0.462 + 0.800i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.01 + 7.51i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-9.17 - 2.45i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + 7.92T + 47T^{2} \)
53 \( 1 + (-2.58 - 2.58i)T + 53iT^{2} \)
59 \( 1 + (4.72 + 1.26i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.60 - 2.78i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (11.5 + 6.64i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.94 + 1.05i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 - 3.44iT - 73T^{2} \)
79 \( 1 + 11.0iT - 79T^{2} \)
83 \( 1 - 2.20T + 83T^{2} \)
89 \( 1 + (-1.59 - 5.96i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (11.8 - 6.83i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.01219679721627826732470923544, −10.48269636140168625913756882636, −9.499921425304570357504516703975, −8.675729766092141646970410012596, −7.82263243765371305492316188890, −6.71478015362547810901355806456, −5.84369468864848863099576441283, −4.03579692871281489286445255973, −2.80262817169839905041252031724, −1.73908083344788817600136477820, 1.24583809940017975522128328297, 2.80522247996991309674877062007, 4.49565379833326418959625650043, 5.53766689974829199838188244677, 6.63375201868049375962458790067, 7.82275704247602431620902000324, 8.583953336183897249917823269706, 9.331515239776040691629157756479, 10.11647966098812629208454388916, 11.02136352523829554670154863466

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