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Properties

Label 2-390-39.5-c1-0-10
Degree $2$
Conductor $390$
Sign $0.997 + 0.0732i$
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 + (0.292 − 1.70i)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 + (−0.292 + 1.70i)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 + (0.119 − 0.696i)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 + (−0.0756 + 0.440i)15-s − 0.250·16-s + 0.342·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0732i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0732i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.997 + 0.0732i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.997 + 0.0732i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59472 - 0.0584830i\)
\(L(\frac12)\) \(\approx\) \(1.59472 - 0.0584830i\)
\(L(\frac{3}{2})\) 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

−11.50364811454564299489628071326, −10.93131159631148556807031474602, −9.021155151199791170859840503911, −8.363162985115952666932544065208, −7.55257768606524366706760087608, −6.23390578994260495613547511873, −5.78129055427179953965725591339, −4.64259234660912567148835955648, −3.18681555762922422595595509476, −1.28122852765509102806085250010, 1.48187566224242678566983793566, 3.61449116945841644620462352411, 4.23245917246151748589438039087, 5.15916527899641744449944928843, 6.49977554905428867559751523115, 7.31448936927655334657759213355, 8.945406334705812799810930837197, 9.674167814606515977652943151133, 10.80749955311754702146434771750, 11.15999334653085802959432225128

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