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Properties

Label 2-390-13.12-c1-0-1
Degree $2$
Conductor $390$
Sign $-0.554 - 0.832i$
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 − i·8-s + 9-s + 10-s + 6i·11-s + 12-s + (2 + 3i)13-s + i·15-s + 16-s − 6·17-s + i·18-s + 6i·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577·3-s − 0.5·4-s − 0.447i·5-s − 0.408i·6-s − 0.353i·8-s + 0.333·9-s + 0.316·10-s + 1.80i·11-s + 0.288·12-s + (0.554 + 0.832i)13-s + 0.258i·15-s + 0.250·16-s − 1.45·17-s + 0.235i·18-s + 1.37i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.554 - 0.832i$
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: \((2,\ 390,\ (\ :1/2),\ -0.554 - 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.417272 + 0.779681i\)
\(L(\frac12)\) \(\approx\) \(0.417272 + 0.779681i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - 3i)T \)
good7 \( 1 - 7T^{2} \)
11 \( 1 - 6iT - 11T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 - 6iT - 19T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 6iT - 37T^{2} \)
41 \( 1 - 12iT - 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 12T + 53T^{2} \)
59 \( 1 - 6iT - 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 12iT - 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 6iT - 97T^{2} \)
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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

−11.67473753329816295033790452958, −10.71225642601125997293160137163, −9.570261488493185067087574930198, −8.985916161971721830176471897460, −7.68054086441949436393320682019, −6.89589188602877229747962071584, −5.97510070968089255756292951133, −4.77927450958039520366412911690, −4.12837175430273221427712761819, −1.77487630784817462268670922760, 0.65066316315450848116912988464, 2.67274417686297425944440739429, 3.74726591476584684784975998567, 5.13576498861079301994618776862, 6.07820729729294637334305501941, 7.12958360839144495513566984949, 8.532505718045756660847517170288, 9.149834721719418994224050663798, 10.54902778668979308990415699076, 11.15067037348505032765538029330

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