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Properties

Label 2-390-65.33-c1-0-12
Degree $2$
Conductor $390$
Sign $0.152 + 0.988i$
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 + (1.61 − 1.54i)5-s + (0.258 − 0.965i)6-s + (1.65 − 0.954i)7-s − 0.999·8-s + (0.866 − 0.499i)9-s + (−0.531 − 2.17i)10-s + (0.562 + 2.09i)11-s + (−0.707 − 0.707i)12-s + (−2.75 + 2.33i)13-s − 1.90i·14-s + (1.16 − 1.91i)15-s + (−0.5 + 0.866i)16-s + (−0.160 + 0.597i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.557 − 0.149i)3-s + (−0.249 − 0.433i)4-s + (0.722 − 0.691i)5-s + (0.105 − 0.394i)6-s + (0.625 − 0.360i)7-s − 0.353·8-s + (0.288 − 0.166i)9-s + (−0.168 − 0.686i)10-s + (0.169 + 0.633i)11-s + (−0.204 − 0.204i)12-s + (−0.762 + 0.646i)13-s − 0.510i·14-s + (0.299 − 0.493i)15-s + (−0.125 + 0.216i)16-s + (−0.0388 + 0.144i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.152 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.152 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.152 + 0.988i$
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: \((2,\ 390,\ (\ :1/2),\ 0.152 + 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62231 - 1.39058i\)
\(L(\frac12)\) \(\approx\) \(1.62231 - 1.39058i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.61 + 1.54i)T \)
13 \( 1 + (2.75 - 2.33i)T \)
good7 \( 1 + (-1.65 + 0.954i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.562 - 2.09i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.160 - 0.597i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (3.46 + 0.927i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.175 + 0.653i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-2.93 - 1.69i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.691 - 0.691i)T + 31iT^{2} \)
37 \( 1 + (2.47 + 1.42i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-7.83 + 2.09i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (3.36 + 0.901i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 - 10.8iT - 47T^{2} \)
53 \( 1 + (1.54 + 1.54i)T + 53iT^{2} \)
59 \( 1 + (3.75 - 14.0i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.85 + 4.94i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.92 + 5.07i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.78 - 6.64i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 - 2.15T + 73T^{2} \)
79 \( 1 - 5.11iT - 79T^{2} \)
83 \( 1 + 3.84iT - 83T^{2} \)
89 \( 1 + (0.804 - 0.215i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (9.16 + 15.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.10380568501098524323868378761, −10.14777436457224558622675058598, −9.356758930722228101397551155580, −8.587912971606644844795734807984, −7.41300356727829449079191611223, −6.24378675183511045139177893594, −4.85460603596794037827704944608, −4.26246339392906625794807980048, −2.50458122809701058903482478905, −1.49903699011818492176589273836, 2.24136175936654680691132963896, 3.36407367923220526732864511513, 4.79600544773271047015848383348, 5.77958303072339976171162647834, 6.74473509450578747108392378615, 7.82636444311510559139021853210, 8.594827441521858494237676277674, 9.625666520123448282544040298753, 10.50109323526909104707495400586, 11.51461584757929491769827498812

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