Properties

Label 2-390-15.2-c1-0-0
Degree $2$
Conductor $390$
Sign $-0.565 - 0.824i$
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 + (−0.807 − 1.53i)3-s − 1.00i·4-s + (−2.02 − 0.947i)5-s + (1.65 + 0.512i)6-s + (−1.59 − 1.59i)7-s + (0.707 + 0.707i)8-s + (−1.69 + 2.47i)9-s + (2.10 − 0.762i)10-s + 2.58i·11-s + (−1.53 + 0.807i)12-s + (0.707 − 0.707i)13-s + 2.25·14-s + (0.183 + 3.86i)15-s − 1.00·16-s + (−0.155 + 0.155i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.465 − 0.884i)3-s − 0.500i·4-s + (−0.905 − 0.423i)5-s + (0.675 + 0.209i)6-s + (−0.603 − 0.603i)7-s + (0.250 + 0.250i)8-s + (−0.565 + 0.824i)9-s + (0.664 − 0.241i)10-s + 0.780i·11-s + (−0.442 + 0.232i)12-s + (0.196 − 0.196i)13-s + 0.603·14-s + (0.0472 + 0.998i)15-s − 0.250·16-s + (−0.0376 + 0.0376i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.565 - 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.565 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.565 - 0.824i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ -0.565 - 0.824i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0804226 + 0.152621i\)
\(L(\frac12)\) \(\approx\) \(0.0804226 + 0.152621i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.807 + 1.53i)T \)
5 \( 1 + (2.02 + 0.947i)T \)
13 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 + (1.59 + 1.59i)T + 7iT^{2} \)
11 \( 1 - 2.58iT - 11T^{2} \)
17 \( 1 + (0.155 - 0.155i)T - 17iT^{2} \)
19 \( 1 - 6.77iT - 19T^{2} \)
23 \( 1 + (-1.81 - 1.81i)T + 23iT^{2} \)
29 \( 1 + 0.890T + 29T^{2} \)
31 \( 1 + 9.39T + 31T^{2} \)
37 \( 1 + (2.88 + 2.88i)T + 37iT^{2} \)
41 \( 1 - 2.11iT - 41T^{2} \)
43 \( 1 + (4.38 - 4.38i)T - 43iT^{2} \)
47 \( 1 + (3.71 - 3.71i)T - 47iT^{2} \)
53 \( 1 + (2.24 + 2.24i)T + 53iT^{2} \)
59 \( 1 - 2.75T + 59T^{2} \)
61 \( 1 + 2.28T + 61T^{2} \)
67 \( 1 + (10.9 + 10.9i)T + 67iT^{2} \)
71 \( 1 - 9.60iT - 71T^{2} \)
73 \( 1 + (1.65 - 1.65i)T - 73iT^{2} \)
79 \( 1 - 12.1iT - 79T^{2} \)
83 \( 1 + (5.93 + 5.93i)T + 83iT^{2} \)
89 \( 1 - 6.43T + 89T^{2} \)
97 \( 1 + (-8.84 - 8.84i)T + 97iT^{2} \)
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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

−11.62697100050132940690560892578, −10.74803502391919228086021390542, −9.779327539752921379141792995392, −8.609773389368522906352473456816, −7.65991757227827502053604181831, −7.20224568931207940999884027899, −6.13516583148325446550920863279, −5.02840466200727436257955196044, −3.63735793942549410888986211273, −1.50899237071781205411836436699, 0.14580006290439120433250589057, 2.89181918755306125328242085941, 3.68818095388681332962841681236, 4.94397686305750377789556549793, 6.25523265763313153540849497054, 7.25384266899798295426802536601, 8.738394541153570946311557395873, 9.051318932764702860741872396519, 10.27880629227957280937143731346, 11.04578298390662721771613128051

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