Properties

Label 2-390-65.37-c1-0-1
Degree $2$
Conductor $390$
Sign $0.352 - 0.935i$
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 + (−2.13 + 0.662i)5-s + (0.965 − 0.258i)6-s + (0.648 − 1.12i)7-s + 0.999i·8-s + (0.866 + 0.499i)9-s + (1.51 − 1.64i)10-s + (2.28 + 0.611i)11-s + (−0.707 + 0.707i)12-s + (−1.97 − 3.01i)13-s + 1.29i·14-s + (2.23 − 0.0876i)15-s + (−0.5 − 0.866i)16-s + (1.79 + 6.70i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.557 − 0.149i)3-s + (0.249 − 0.433i)4-s + (−0.955 + 0.296i)5-s + (0.394 − 0.105i)6-s + (0.245 − 0.424i)7-s + 0.353i·8-s + (0.288 + 0.166i)9-s + (0.480 − 0.519i)10-s + (0.688 + 0.184i)11-s + (−0.204 + 0.204i)12-s + (−0.547 − 0.836i)13-s + 0.346i·14-s + (0.576 − 0.0226i)15-s + (−0.125 − 0.216i)16-s + (0.435 + 1.62i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.352 - 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.352 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.352 - 0.935i$
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: \((2,\ 390,\ (\ :1/2),\ 0.352 - 0.935i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.550301 + 0.380836i\)
\(L(\frac12)\) \(\approx\) \(0.550301 + 0.380836i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.13 - 0.662i)T \)
13 \( 1 + (1.97 + 3.01i)T \)
good7 \( 1 + (-0.648 + 1.12i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.28 - 0.611i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.79 - 6.70i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.22 - 4.55i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.31 - 4.91i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-5.49 + 3.16i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.24 - 6.24i)T + 31iT^{2} \)
37 \( 1 + (-1.61 - 2.80i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.19 - 8.17i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (0.778 - 0.208i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + 11.6T + 47T^{2} \)
53 \( 1 + (-8.79 + 8.79i)T - 53iT^{2} \)
59 \( 1 + (-5.81 + 1.55i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-2.10 + 3.64i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.573 + 0.331i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (12.2 - 3.28i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 - 5.04iT - 73T^{2} \)
79 \( 1 + 13.2iT - 79T^{2} \)
83 \( 1 + 2.04T + 83T^{2} \)
89 \( 1 + (-1.16 + 4.33i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-5.19 - 2.99i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.56032337359639475421996440681, −10.31777817735671273585367534382, −10.04451766212725840603212697637, −8.296138281720920951206491358839, −7.919280697608496303742528542993, −6.87020623937319029339407121705, −5.99234927567993588250673502986, −4.67501766534195048702923303575, −3.44844892837202402036542526138, −1.28310884254163375514355069167, 0.68547539238969497380705407674, 2.68330593609035425353406323378, 4.20999929765425586298202143421, 5.06944970740880090326960491396, 6.66787334131761456195288627876, 7.38895744170656864269009005704, 8.599563821229286143435491045598, 9.243744556612418660746947188202, 10.24418892571744004551100567144, 11.49842018946703566575612757589

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