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Properties

Label 2-390-15.8-c1-0-16
Degree $2$
Conductor $390$
Sign $0.891 - 0.453i$
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 + (−1.45 + 0.932i)3-s + 1.00i·4-s + (2.00 − 0.984i)5-s + (−1.69 − 0.372i)6-s + (3.42 − 3.42i)7-s + (−0.707 + 0.707i)8-s + (1.25 − 2.72i)9-s + (2.11 + 0.723i)10-s − 1.31i·11-s + (−0.932 − 1.45i)12-s + (−0.707 − 0.707i)13-s + 4.84·14-s + (−2.01 + 3.30i)15-s − 1.00·16-s + (2.63 + 2.63i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.842 + 0.538i)3-s + 0.500i·4-s + (0.897 − 0.440i)5-s + (−0.690 − 0.151i)6-s + (1.29 − 1.29i)7-s + (−0.250 + 0.250i)8-s + (0.419 − 0.907i)9-s + (0.669 + 0.228i)10-s − 0.395i·11-s + (−0.269 − 0.421i)12-s + (−0.196 − 0.196i)13-s + 1.29·14-s + (−0.519 + 0.854i)15-s − 0.250·16-s + (0.638 + 0.638i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 - 0.453i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.891 - 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.891 - 0.453i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.891 - 0.453i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69102 + 0.405918i\)
\(L(\frac12)\) \(\approx\) \(1.69102 + 0.405918i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.45 - 0.932i)T \)
5 \( 1 + (-2.00 + 0.984i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + (-3.42 + 3.42i)T - 7iT^{2} \)
11 \( 1 + 1.31iT - 11T^{2} \)
17 \( 1 + (-2.63 - 2.63i)T + 17iT^{2} \)
19 \( 1 + 0.405iT - 19T^{2} \)
23 \( 1 + (5.29 - 5.29i)T - 23iT^{2} \)
29 \( 1 - 4.56T + 29T^{2} \)
31 \( 1 + 2.96T + 31T^{2} \)
37 \( 1 + (4.56 - 4.56i)T - 37iT^{2} \)
41 \( 1 - 1.94iT - 41T^{2} \)
43 \( 1 + (-3.52 - 3.52i)T + 43iT^{2} \)
47 \( 1 + (3.58 + 3.58i)T + 47iT^{2} \)
53 \( 1 + (-0.235 + 0.235i)T - 53iT^{2} \)
59 \( 1 + 11.5T + 59T^{2} \)
61 \( 1 - 10.5T + 61T^{2} \)
67 \( 1 + (6.22 - 6.22i)T - 67iT^{2} \)
71 \( 1 + 10.7iT - 71T^{2} \)
73 \( 1 + (-9.20 - 9.20i)T + 73iT^{2} \)
79 \( 1 + 11.4iT - 79T^{2} \)
83 \( 1 + (11.8 - 11.8i)T - 83iT^{2} \)
89 \( 1 + 4.30T + 89T^{2} \)
97 \( 1 + (0.357 - 0.357i)T - 97iT^{2} \)
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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

−11.37127282306409088085219070943, −10.48008518825735075583726649170, −9.819912056175531698299242716576, −8.468695909223377693248302957759, −7.54082844650510829978052484395, −6.36949811931635105809397996317, −5.44162814345766822257175388760, −4.72861560915849391033786335376, −3.73490439688983589745153231829, −1.36723524295103406347382012671, 1.72663653907818813717023156072, 2.50274511472738461306388546402, 4.65894718531443665547359824687, 5.43932542132293547643298720840, 6.13011899953840864294190249476, 7.29420451516688973425504975875, 8.508445485887628747971434572279, 9.699357969904168703386938503086, 10.61341644228495745024700420482, 11.37226265574748092323536121970

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