Properties

Label 2-390-65.49-c1-0-8
Degree $2$
Conductor $390$
Sign $0.964 - 0.263i$
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 + (2.10 − 0.767i)5-s + (0.866 + 0.499i)6-s + (0.823 − 1.42i)7-s − 0.999·8-s + (0.499 − 0.866i)9-s + (1.71 + 1.43i)10-s + (2.08 − 1.20i)11-s + 0.999i·12-s + (−3.59 − 0.256i)13-s + 1.64·14-s + (1.43 − 1.71i)15-s + (−0.5 − 0.866i)16-s + (0.210 + 0.121i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.499 − 0.288i)3-s + (−0.249 + 0.433i)4-s + (0.939 − 0.343i)5-s + (0.353 + 0.204i)6-s + (0.311 − 0.538i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (0.542 + 0.453i)10-s + (0.628 − 0.362i)11-s + 0.288i·12-s + (−0.997 − 0.0710i)13-s + 0.439·14-s + (0.370 − 0.442i)15-s + (−0.125 − 0.216i)16-s + (0.0511 + 0.0295i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.263i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 - 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.964 - 0.263i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.964 - 0.263i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.13054 + 0.285623i\)
\(L(\frac12)\) \(\approx\) \(2.13054 + 0.285623i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.10 + 0.767i)T \)
13 \( 1 + (3.59 + 0.256i)T \)
good7 \( 1 + (-0.823 + 1.42i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.08 + 1.20i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.210 - 0.121i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.82 - 2.20i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (7.46 - 4.31i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.0221 - 0.0383i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.24iT - 31T^{2} \)
37 \( 1 + (-4.47 - 7.74i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.210 + 0.121i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.82 + 3.36i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 7.29T + 47T^{2} \)
53 \( 1 + 2.44iT - 53T^{2} \)
59 \( 1 + (8.35 + 4.82i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.31 + 2.27i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.937 - 1.62i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.53 + 3.77i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 1.70T + 73T^{2} \)
79 \( 1 - 6.79T + 79T^{2} \)
83 \( 1 + 17.4T + 83T^{2} \)
89 \( 1 + (8.69 - 5.02i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.25 + 14.3i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.62219330583130746539590382585, −10.07134353476214695810557071579, −9.542600429727680382266758355759, −8.404735647343583612549044810447, −7.61283217624142933396162888334, −6.60130266394013001422650938651, −5.63040935282401781438297419034, −4.56528479993305567983187964373, −3.24811105563705153982891218446, −1.61762234901410345004732574022, 1.93572629101447205079479827539, 2.79056651286033235251908450162, 4.25251891306062125160942206767, 5.28536022828513105921818933046, 6.32803111569515528256188791019, 7.57325241383049770654442987754, 8.873515774353286620284982667740, 9.666883534743682816600528166083, 10.15316379250888785811195429203, 11.34459080059746005741795664143

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