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Properties

Label 2-390-65.33-c1-0-11
Degree $2$
Conductor $390$
Sign $0.959 + 0.282i$
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  + (−0.5 + 0.866i)2-s + (0.965 − 0.258i)3-s + (−0.499 − 0.866i)4-s + (1.83 − 1.27i)5-s + (−0.258 + 0.965i)6-s + (0.838 − 0.484i)7-s + 0.999·8-s + (0.866 − 0.499i)9-s + (0.187 + 2.22i)10-s + (−1.30 − 4.85i)11-s + (−0.707 − 0.707i)12-s + (−1.46 − 3.29i)13-s + 0.968i·14-s + (1.44 − 1.70i)15-s + (−0.5 + 0.866i)16-s + (−1.82 + 6.79i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.557 − 0.149i)3-s + (−0.249 − 0.433i)4-s + (0.821 − 0.570i)5-s + (−0.105 + 0.394i)6-s + (0.316 − 0.183i)7-s + 0.353·8-s + (0.288 − 0.166i)9-s + (0.0593 + 0.704i)10-s + (−0.391 − 1.46i)11-s + (−0.204 − 0.204i)12-s + (−0.406 − 0.913i)13-s + 0.258i·14-s + (0.372 − 0.441i)15-s + (−0.125 + 0.216i)16-s + (−0.441 + 1.64i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.282i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 + 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.959 + 0.282i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.959 + 0.282i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49409 - 0.215810i\)
\(L(\frac12)\) \(\approx\) \(1.49409 - 0.215810i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 + (-1.83 + 1.27i)T \)
13 \( 1 + (1.46 + 3.29i)T \)
good7 \( 1 + (-0.838 + 0.484i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.30 + 4.85i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.82 - 6.79i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-2.91 - 0.780i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.01 + 3.80i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (3.52 + 2.03i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-6.94 - 6.94i)T + 31iT^{2} \)
37 \( 1 + (-7.90 - 4.56i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (7.00 - 1.87i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-9.01 - 2.41i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 - 10.2iT - 47T^{2} \)
53 \( 1 + (-4.84 - 4.84i)T + 53iT^{2} \)
59 \( 1 + (-1.32 + 4.96i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (6.98 + 12.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.94 - 3.37i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.745 - 2.78i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + 5.28T + 73T^{2} \)
79 \( 1 + 8.53iT - 79T^{2} \)
83 \( 1 - 7.83iT - 83T^{2} \)
89 \( 1 + (9.75 - 2.61i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-5.23 - 9.05i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.92732162973281272944637998104, −10.22853649786585551664745352193, −9.262530265351165392762925478742, −8.222504134957092180960994375815, −8.034593796151167184823377419120, −6.38072077481849329107134448274, −5.72442467827787250690000562845, −4.52137202447027917765999642449, −2.88012253445099569915310700109, −1.19587031273990835778984216071, 1.98784755598132688091964813708, 2.68571160590401864267149894716, 4.29089712828373558317965452718, 5.33072873285549592878100680729, 7.04361421206881149174185113230, 7.51276548794469989874349243681, 9.031483387970023445467416480138, 9.617647736414011135386862806920, 10.14176274184850324833188933118, 11.39415915621175591924777057460

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