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Properties

Label 2-390-195.59-c1-0-23
Degree $2$
Conductor $390$
Sign $-0.283 + 0.958i$
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 + (−1.52 − 0.823i)3-s + (0.866 − 0.499i)4-s + (1.09 − 1.94i)5-s + (−1.68 − 0.400i)6-s + (0.391 − 1.46i)7-s + (0.707 − 0.707i)8-s + (1.64 + 2.50i)9-s + (0.555 − 2.16i)10-s + (−1.75 + 0.468i)11-s + (−1.73 + 0.0488i)12-s + (−2.94 − 2.07i)13-s − 1.51i·14-s + (−3.27 + 2.06i)15-s + (0.500 − 0.866i)16-s + (0.174 − 0.100i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (−0.879 − 0.475i)3-s + (0.433 − 0.249i)4-s + (0.490 − 0.871i)5-s + (−0.687 − 0.163i)6-s + (0.147 − 0.552i)7-s + (0.249 − 0.249i)8-s + (0.547 + 0.836i)9-s + (0.175 − 0.684i)10-s + (−0.527 + 0.141i)11-s + (−0.499 + 0.0140i)12-s + (−0.817 − 0.576i)13-s − 0.404i·14-s + (−0.845 + 0.533i)15-s + (0.125 − 0.216i)16-s + (0.0422 − 0.0243i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.283 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.283 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.283 + 0.958i$
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: \((2,\ 390,\ (\ :1/2),\ -0.283 + 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.928442 - 1.24296i\)
\(L(\frac12)\) \(\approx\) \(0.928442 - 1.24296i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.52 + 0.823i)T \)
5 \( 1 + (-1.09 + 1.94i)T \)
13 \( 1 + (2.94 + 2.07i)T \)
good7 \( 1 + (-0.391 + 1.46i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (1.75 - 0.468i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-0.174 + 0.100i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.975 + 3.64i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.36 - 0.785i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.92 - 3.99i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (6.25 + 6.25i)T + 31iT^{2} \)
37 \( 1 + (-4.71 + 1.26i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-2.38 - 8.89i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-1.71 - 2.96i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.03 - 6.03i)T - 47iT^{2} \)
53 \( 1 - 12.5T + 53T^{2} \)
59 \( 1 + (-0.00595 + 0.0222i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (0.554 + 0.959i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.78 - 6.65i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (7.88 + 2.11i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.794 - 0.794i)T + 73iT^{2} \)
79 \( 1 - 9.56T + 79T^{2} \)
83 \( 1 + (-9.32 - 9.32i)T + 83iT^{2} \)
89 \( 1 + (3.65 - 0.980i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-13.4 - 3.60i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.19076317437693470346721627948, −10.35859123176551786191275371830, −9.483529392391406870525063391559, −7.965168772226569247701993144808, −7.15278786012263350169751431697, −5.99533328822180827449360014025, −5.11358888004370555217602931288, −4.47753725052342045800281042169, −2.51669712291653276765789857772, −0.954401224479591655979602908635, 2.29179006759118408266539872434, 3.64899853192910694624842128005, 4.97562695338265772459027349380, 5.70731053960865625627923510037, 6.60297364606044775008267631299, 7.47486636595775741143576946703, 8.984809150459244408958662037360, 10.13683752194951616019155468674, 10.64765434582622112975845190456, 11.75495038566764639634409260724

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