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Properties

Label 2-390-65.28-c1-0-3
Degree $2$
Conductor $390$
Sign $-0.324 - 0.945i$
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 + (−0.258 + 0.965i)3-s + (0.499 − 0.866i)4-s + (−2.12 + 0.707i)5-s + (0.258 + 0.965i)6-s + (−1.88 + 3.26i)7-s − 0.999i·8-s + (−0.866 − 0.499i)9-s + (−1.48 + 1.67i)10-s + (−1.23 + 4.60i)11-s + (0.707 + 0.707i)12-s + (−3.53 − 0.707i)13-s + 3.76i·14-s + (−0.133 − 2.23i)15-s + (−0.5 − 0.866i)16-s + (6.22 − 1.66i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.149 + 0.557i)3-s + (0.249 − 0.433i)4-s + (−0.948 + 0.316i)5-s + (0.105 + 0.394i)6-s + (−0.711 + 1.23i)7-s − 0.353i·8-s + (−0.288 − 0.166i)9-s + (−0.469 + 0.529i)10-s + (−0.372 + 1.38i)11-s + (0.204 + 0.204i)12-s + (−0.980 − 0.196i)13-s + 1.00i·14-s + (−0.0345 − 0.576i)15-s + (−0.125 − 0.216i)16-s + (1.51 − 0.404i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.324 - 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.324 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.324 - 0.945i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ -0.324 - 0.945i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.602269 + 0.843258i\)
\(L(\frac12)\) \(\approx\) \(0.602269 + 0.843258i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.258 - 0.965i)T \)
5 \( 1 + (2.12 - 0.707i)T \)
13 \( 1 + (3.53 + 0.707i)T \)
good7 \( 1 + (1.88 - 3.26i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.23 - 4.60i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-6.22 + 1.66i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (5.43 - 1.45i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-7.94 - 2.12i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (4.61 - 2.66i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.648 - 0.648i)T - 31iT^{2} \)
37 \( 1 + (-0.423 - 0.733i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.23 - 0.598i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-0.155 - 0.579i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + 1.59T + 47T^{2} \)
53 \( 1 + (-4.92 - 4.92i)T + 53iT^{2} \)
59 \( 1 + (2.86 + 10.7i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.30 - 2.26i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.210 + 0.121i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.87 - 10.7i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + 8.62iT - 73T^{2} \)
79 \( 1 - 11.7iT - 79T^{2} \)
83 \( 1 - 14.8T + 83T^{2} \)
89 \( 1 + (2.58 + 0.691i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-6.60 - 3.81i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.77335179551455105223322488702, −10.75106685863279565560819946563, −9.878265992964300333348350826265, −9.141287089957026469282272680569, −7.73522212177806922942085320906, −6.81573061837245057757596255173, −5.48454642667563511331476961108, −4.74242394776852288219076691041, −3.44733099803808610025648068257, −2.52147052455716793953828149440, 0.55886571739866046879485687441, 3.05059978159542802998633991509, 3.97701724763419902481156047237, 5.17739428814890195762831522638, 6.36285350221058473465652555352, 7.29489192170143673126528275971, 7.88018819673516563545146091354, 8.946308869374130309348735418639, 10.44795730075208321172679620677, 11.14515959723353320254239924726

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