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Properties

Label 2-390-13.4-c1-0-6
Degree $2$
Conductor $390$
Sign $0.964 + 0.265i$
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 + (0.5 − 0.866i)3-s + (0.499 + 0.866i)4-s − i·5-s + (0.866 − 0.499i)6-s + (1.73 − i)7-s + 0.999i·8-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)10-s + (0.401 + 0.232i)11-s + 0.999·12-s + (1 − 3.46i)13-s + 1.99·14-s + (−0.866 − 0.5i)15-s + (−0.5 + 0.866i)16-s + (2 + 3.46i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.288 − 0.499i)3-s + (0.249 + 0.433i)4-s − 0.447i·5-s + (0.353 − 0.204i)6-s + (0.654 − 0.377i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (0.158 − 0.273i)10-s + (0.121 + 0.0699i)11-s + 0.288·12-s + (0.277 − 0.960i)13-s + 0.534·14-s + (−0.223 − 0.129i)15-s + (−0.125 + 0.216i)16-s + (0.485 + 0.840i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.964 + 0.265i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.964 + 0.265i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.17449 - 0.293372i\)
\(L(\frac12)\) \(\approx\) \(2.17449 - 0.293372i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + iT \)
13 \( 1 + (-1 + 3.46i)T \)
good7 \( 1 + (-1.73 + i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.401 - 0.232i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.464 - 0.267i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.133 - 0.232i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.86 - 3.23i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.73iT - 31T^{2} \)
37 \( 1 + (-1.03 - 0.598i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.73 + i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.964 - 1.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 10.4iT - 47T^{2} \)
53 \( 1 + 12.9T + 53T^{2} \)
59 \( 1 + (1.33 - 0.767i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.19 + 9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.92 - 2.26i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.26 - 4.19i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 + 0.0717T + 79T^{2} \)
83 \( 1 - 4.92iT - 83T^{2} \)
89 \( 1 + (-6.46 - 3.73i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.46 - 3.73i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.40015385130113339053797071741, −10.55542212491616481116556327536, −9.257212128458081610940022676657, −8.078373952676704552512322283425, −7.75338668018561427806800200379, −6.43490527312637545586264229066, −5.49456691732621386830599719431, −4.38025457321253419724202458380, −3.18842569997507225046348647153, −1.49669044634574517470979148698, 1.96770013224055879699421583161, 3.22727748026356694840297128846, 4.35518390664800572302136329605, 5.30852660327261309616696853991, 6.45520172121432949839163273474, 7.58993471480117470236079886203, 8.765550053429603303729873530969, 9.632955798680517045224361016152, 10.58482941794237321934998320691, 11.49662851803774000723928917146

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