Properties

Label 2-390-13.12-c1-0-6
Degree $2$
Conductor $390$
Sign $1$
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  + i·2-s − 3-s − 4-s + i·5-s i·6-s − 4.60i·7-s i·8-s + 9-s − 10-s + 12-s + 3.60·13-s + 4.60·14-s i·15-s + 16-s + 4.60·17-s + i·18-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577·3-s − 0.5·4-s + 0.447i·5-s − 0.408i·6-s − 1.74i·7-s − 0.353i·8-s + 0.333·9-s − 0.316·10-s + 0.288·12-s + 1.00·13-s + 1.23·14-s − 0.258i·15-s + 0.250·16-s + 1.11·17-s + 0.235i·18-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $1$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08592\)
\(L(\frac12)\) \(\approx\) \(1.08592\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + T \)
5 \( 1 - iT \)
13 \( 1 - 3.60T \)
good7 \( 1 + 4.60iT - 7T^{2} \)
11 \( 1 - 11T^{2} \)
17 \( 1 - 4.60T + 17T^{2} \)
19 \( 1 + 4.60iT - 19T^{2} \)
23 \( 1 + 1.39T + 23T^{2} \)
29 \( 1 - 4.60T + 29T^{2} \)
31 \( 1 - 6iT - 31T^{2} \)
37 \( 1 + 9.21iT - 37T^{2} \)
41 \( 1 - 3.21iT - 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + 9.21iT - 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 9.21iT - 59T^{2} \)
61 \( 1 + 11.2T + 61T^{2} \)
67 \( 1 + 3.21iT - 67T^{2} \)
71 \( 1 - 9.21iT - 71T^{2} \)
73 \( 1 + 1.39iT - 73T^{2} \)
79 \( 1 + 14.4T + 79T^{2} \)
83 \( 1 - 2.78iT - 83T^{2} \)
89 \( 1 - 15.2iT - 89T^{2} \)
97 \( 1 - 1.39iT - 97T^{2} \)
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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

−10.98590480735476491680464623781, −10.55545631564459147692087046345, −9.609856461462203484343119786176, −8.292324832698794270196394052586, −7.28408370850552558868117719050, −6.75684921745809903690553621907, −5.67092512818433186732684878306, −4.44528630915969865537365404958, −3.49828169818915795469979154788, −0.913082115461151326886198426371, 1.47051304109435739982787953074, 2.95706065298897152912152530358, 4.36571903843325057029973807942, 5.66855613071576167916315318375, 6.00583179444314314035526684452, 7.911395839490672226612383495233, 8.701653561132389674737517724002, 9.571349330763240661057203973055, 10.46115682256250730965007588703, 11.58732145091931953414428592509

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