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Properties

Label 2-390-195.77-c1-0-7
Degree $2$
Conductor $390$
Sign $-0.337 - 0.941i$
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 + (−1.72 + 0.128i)3-s + 1.00i·4-s + (1.56 + 1.60i)5-s + (−1.31 − 1.13i)6-s + (0.236 − 0.236i)7-s + (−0.707 + 0.707i)8-s + (2.96 − 0.445i)9-s + (−0.0281 + 2.23i)10-s + 2.01·11-s + (−0.128 − 1.72i)12-s + (−0.965 + 3.47i)13-s + 0.334·14-s + (−2.90 − 2.56i)15-s − 1.00·16-s + (−2.69 + 2.69i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.997 + 0.0743i)3-s + 0.500i·4-s + (0.698 + 0.715i)5-s + (−0.535 − 0.461i)6-s + (0.0893 − 0.0893i)7-s + (−0.250 + 0.250i)8-s + (0.988 − 0.148i)9-s + (−0.00888 + 0.707i)10-s + 0.606·11-s + (−0.0371 − 0.498i)12-s + (−0.267 + 0.963i)13-s + 0.0893·14-s + (−0.749 − 0.662i)15-s − 0.250·16-s + (−0.652 + 0.652i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.337 - 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.337 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.337 - 0.941i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ -0.337 - 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.793635 + 1.12771i\)
\(L(\frac12)\) \(\approx\) \(0.793635 + 1.12771i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.72 - 0.128i)T \)
5 \( 1 + (-1.56 - 1.60i)T \)
13 \( 1 + (0.965 - 3.47i)T \)
good7 \( 1 + (-0.236 + 0.236i)T - 7iT^{2} \)
11 \( 1 - 2.01T + 11T^{2} \)
17 \( 1 + (2.69 - 2.69i)T - 17iT^{2} \)
19 \( 1 + 8.01T + 19T^{2} \)
23 \( 1 + (-6.23 - 6.23i)T + 23iT^{2} \)
29 \( 1 - 4.55T + 29T^{2} \)
31 \( 1 + 7.05iT - 31T^{2} \)
37 \( 1 + (-1.19 + 1.19i)T - 37iT^{2} \)
41 \( 1 - 7.90T + 41T^{2} \)
43 \( 1 + (-2.76 + 2.76i)T - 43iT^{2} \)
47 \( 1 + (2.04 + 2.04i)T + 47iT^{2} \)
53 \( 1 + (6.83 + 6.83i)T + 53iT^{2} \)
59 \( 1 + 12.3iT - 59T^{2} \)
61 \( 1 - 5.48T + 61T^{2} \)
67 \( 1 + (4.26 - 4.26i)T - 67iT^{2} \)
71 \( 1 - 9.88T + 71T^{2} \)
73 \( 1 + (-2.59 - 2.59i)T + 73iT^{2} \)
79 \( 1 + 10.6iT - 79T^{2} \)
83 \( 1 + (-2.98 + 2.98i)T - 83iT^{2} \)
89 \( 1 + 1.02iT - 89T^{2} \)
97 \( 1 + (-10.9 + 10.9i)T - 97iT^{2} \)
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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

−11.34678798956324135224319448916, −11.02946040180827812527178521020, −9.836024462894124390022276416600, −8.948096668846804080579265685897, −7.41839771099067284970377197674, −6.50589064055154659254809449926, −6.13216759699674179985559862275, −4.82761119384615477769411853213, −3.89419276402662869630605036076, −2.02522340075283000662721089601, 0.926117025711164724805246347381, 2.48136402445213401696885200472, 4.43264975702110278365587076159, 4.98671415776128186694385591806, 6.11696992094226144209880096765, 6.80163913319630490678692911086, 8.475792897148642466977243370872, 9.393586978603315736682362531235, 10.50381500107916982044304599508, 10.92461325738800068552799440246

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