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Properties

Label 2-390-65.58-c1-0-1
Degree $2$
Conductor $390$
Sign $0.204 - 0.978i$
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 + (−2.12 + 0.707i)5-s + (−0.965 − 0.258i)6-s + (2.29 + 3.98i)7-s − 0.999i·8-s + (0.866 − 0.499i)9-s + (2.19 + 0.448i)10-s + (−3.47 + 0.930i)11-s + (0.707 + 0.707i)12-s + (−3.53 − 0.707i)13-s − 4.59i·14-s + (−1.86 + 1.23i)15-s + (−0.5 + 0.866i)16-s + (−0.401 + 1.49i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.557 − 0.149i)3-s + (0.249 + 0.433i)4-s + (−0.948 + 0.316i)5-s + (−0.394 − 0.105i)6-s + (0.868 + 1.50i)7-s − 0.353i·8-s + (0.288 − 0.166i)9-s + (0.692 + 0.141i)10-s + (−1.04 + 0.280i)11-s + (0.204 + 0.204i)12-s + (−0.980 − 0.196i)13-s − 1.22i·14-s + (−0.481 + 0.318i)15-s + (−0.125 + 0.216i)16-s + (−0.0973 + 0.363i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.204 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.204 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.204 - 0.978i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.204 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.672151 + 0.546195i\)
\(L(\frac12)\) \(\approx\) \(0.672151 + 0.546195i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.12 - 0.707i)T \)
13 \( 1 + (3.53 + 0.707i)T \)
good7 \( 1 + (-2.29 - 3.98i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.47 - 0.930i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (0.401 - 1.49i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.97 - 7.37i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.298 - 1.11i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-4.97 - 2.87i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.351 - 0.351i)T - 31iT^{2} \)
37 \( 1 + (-5.11 + 8.85i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.93 - 7.22i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-1.38 - 0.369i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 - 6.76T + 47T^{2} \)
53 \( 1 + (8.92 + 8.92i)T + 53iT^{2} \)
59 \( 1 + (-8.74 - 2.34i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.721 - 1.25i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.210 + 0.121i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.94 + 2.12i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 - 3.79iT - 73T^{2} \)
79 \( 1 - 2.76iT - 79T^{2} \)
83 \( 1 + 8.75T + 83T^{2} \)
89 \( 1 + (-0.166 - 0.620i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-4.87 + 2.81i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.51124250316457440710764404608, −10.59747434354532502708187229185, −9.664996500403300969505120493817, −8.456819236102028692890297238807, −8.101811314304864572318875446869, −7.29422356537932989020259512874, −5.73678928347303532196607303490, −4.46046809530004339248908941660, −2.96443353568233532355298971802, −2.06075250906399650600343360328, 0.64710594996441941816302525624, 2.68552850555238078552441781907, 4.37508987666113162499529834501, 4.89950140843960021266587718054, 6.92074168098163918786689067453, 7.57084003739988555337093109773, 8.157260645324619736720026472544, 9.099863354289127668269849443255, 10.27537902480139580643039688247, 10.89564762965244408137340148912

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