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Properties

Label 2-390-39.8-c1-0-9
Degree $2$
Conductor $390$
Sign $-0.263 + 0.964i$
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 + (−1 − 1.41i)3-s − 1.00i·4-s + (0.707 − 0.707i)5-s + (1.70 + 0.292i)6-s + (2 − 2i)7-s + (0.707 + 0.707i)8-s + (−1.00 + 2.82i)9-s + 1.00i·10-s + (−4.24 − 4.24i)11-s + (−1.41 + 1.00i)12-s + (3 + 2i)13-s + 2.82i·14-s + (−1.70 − 0.292i)15-s − 1.00·16-s − 1.41·17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.577 − 0.816i)3-s − 0.500i·4-s + (0.316 − 0.316i)5-s + (0.696 + 0.119i)6-s + (0.755 − 0.755i)7-s + (0.250 + 0.250i)8-s + (−0.333 + 0.942i)9-s + 0.316i·10-s + (−1.27 − 1.27i)11-s + (−0.408 + 0.288i)12-s + (0.832 + 0.554i)13-s + 0.755i·14-s + (−0.440 − 0.0756i)15-s − 0.250·16-s − 0.342·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.263 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.263 + 0.964i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (281, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ -0.263 + 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.459903 - 0.602298i\)
\(L(\frac12)\) \(\approx\) \(0.459903 - 0.602298i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 1.41i)T \)
5 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 + (-3 - 2i)T \)
good7 \( 1 + (-2 + 2i)T - 7iT^{2} \)
11 \( 1 + (4.24 + 4.24i)T + 11iT^{2} \)
17 \( 1 + 1.41T + 17T^{2} \)
19 \( 1 + 19iT^{2} \)
23 \( 1 + 4.24T + 23T^{2} \)
29 \( 1 + 9.89iT - 29T^{2} \)
31 \( 1 + (7 + 7i)T + 31iT^{2} \)
37 \( 1 + (-3 + 3i)T - 37iT^{2} \)
41 \( 1 + (1.41 - 1.41i)T - 41iT^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + (-4.24 - 4.24i)T + 47iT^{2} \)
53 \( 1 - 5.65iT - 53T^{2} \)
59 \( 1 + (-8.48 - 8.48i)T + 59iT^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + (-3 - 3i)T + 67iT^{2} \)
71 \( 1 + (-2.82 + 2.82i)T - 71iT^{2} \)
73 \( 1 - 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (-8.48 + 8.48i)T - 83iT^{2} \)
89 \( 1 + (7.07 + 7.07i)T + 89iT^{2} \)
97 \( 1 + (-8 - 8i)T + 97iT^{2} \)
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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

−11.02716662045344635445178170472, −10.34331659010977287136921900380, −8.948755137902126905091898159084, −7.985667336992610258544414068342, −7.55815401452421260589109160576, −6.12902095355161620761522031492, −5.66861025905782657992801058871, −4.32237017230639587145641179790, −2.11602835823495382303226409215, −0.62798853802220257887368018721, 1.97109278796746556937353066664, 3.35581357064405588198968368215, 4.84344528957901258815161864057, 5.52988762381361528154090112068, 6.91077959633222294199376639543, 8.151527824217868199772553082084, 8.994616916909919245167610318688, 10.01245442307718603735429245999, 10.63212011541805402165490492033, 11.25861704772592081600161796259

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