Properties

Label 2-390-195.59-c1-0-21
Degree $2$
Conductor $390$
Sign $-0.531 + 0.846i$
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  + (−0.965 + 0.258i)2-s + (−0.681 − 1.59i)3-s + (0.866 − 0.499i)4-s + (1.33 − 1.79i)5-s + (1.07 + 1.36i)6-s + (1.22 − 4.57i)7-s + (−0.707 + 0.707i)8-s + (−2.07 + 2.16i)9-s + (−0.821 + 2.07i)10-s + (0.957 − 0.256i)11-s + (−1.38 − 1.03i)12-s + (3.59 + 0.326i)13-s + 4.73i·14-s + (−3.76 − 0.898i)15-s + (0.500 − 0.866i)16-s + (−6.31 + 3.64i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (−0.393 − 0.919i)3-s + (0.433 − 0.249i)4-s + (0.595 − 0.803i)5-s + (0.436 + 0.556i)6-s + (0.463 − 1.72i)7-s + (−0.249 + 0.249i)8-s + (−0.690 + 0.723i)9-s + (−0.259 + 0.657i)10-s + (0.288 − 0.0773i)11-s + (−0.400 − 0.299i)12-s + (0.995 + 0.0906i)13-s + 1.26i·14-s + (−0.972 − 0.231i)15-s + (0.125 − 0.216i)16-s + (−1.53 + 0.884i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.531 + 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.531 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.531 + 0.846i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ -0.531 + 0.846i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.456539 - 0.825904i\)
\(L(\frac12)\) \(\approx\) \(0.456539 - 0.825904i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 + (0.681 + 1.59i)T \)
5 \( 1 + (-1.33 + 1.79i)T \)
13 \( 1 + (-3.59 - 0.326i)T \)
good7 \( 1 + (-1.22 + 4.57i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.957 + 0.256i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (6.31 - 3.64i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.503 + 1.88i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.14 - 1.81i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.96 + 1.70i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.89 - 2.89i)T + 31iT^{2} \)
37 \( 1 + (4.31 - 1.15i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.42 + 5.33i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (3.67 + 6.37i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.710 - 0.710i)T - 47iT^{2} \)
53 \( 1 - 9.89T + 53T^{2} \)
59 \( 1 + (0.358 - 1.33i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-3.47 - 6.02i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.67 - 10.0i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (0.344 + 0.0923i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-8.41 - 8.41i)T + 73iT^{2} \)
79 \( 1 - 0.626T + 79T^{2} \)
83 \( 1 + (-2.20 - 2.20i)T + 83iT^{2} \)
89 \( 1 + (-1.91 + 0.513i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (10.6 + 2.86i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.91514393054906417649427938810, −10.27911565375841980289795021471, −8.851429947389130014022555314678, −8.334235429713307422156108533382, −7.12697901876250504729650021072, −6.57557042795221740038984038792, −5.37488234297855133418991485191, −4.09471140249293936849758656912, −1.82502733365186849216861048720, −0.855017422375750408545534602494, 2.15491871612766527291996393163, 3.25611067674030808425475969264, 4.92556408275378892466621542890, 5.97083481100886438179901362753, 6.67731102263723830846352156289, 8.362656149632517939913497593012, 9.090446531739165461924411584836, 9.655323966590673779076869023770, 10.85444728115713559960250338178, 11.29648854101890110180531901748

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