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Properties

Label 2-390-65.47-c1-0-9
Degree $2$
Conductor $390$
Sign $0.581 + 0.813i$
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 + (−0.707 + 0.707i)3-s − 4-s + (−2.09 − 0.775i)5-s + (−0.707 − 0.707i)6-s − 0.946·7-s − i·8-s − 1.00i·9-s + (0.775 − 2.09i)10-s + (3.13 − 3.13i)11-s + (0.707 − 0.707i)12-s + (−3.59 − 0.250i)13-s − 0.946i·14-s + (2.03 − 0.934i)15-s + 16-s + (3.42 − 3.42i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.408 + 0.408i)3-s − 0.5·4-s + (−0.937 − 0.346i)5-s + (−0.288 − 0.288i)6-s − 0.357·7-s − 0.353i·8-s − 0.333i·9-s + (0.245 − 0.663i)10-s + (0.944 − 0.944i)11-s + (0.204 − 0.204i)12-s + (−0.997 − 0.0696i)13-s − 0.253i·14-s + (0.524 − 0.241i)15-s + 0.250·16-s + (0.829 − 0.829i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.581 + 0.813i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.581 + 0.813i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.510463 - 0.262396i\)
\(L(\frac12)\) \(\approx\) \(0.510463 - 0.262396i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (2.09 + 0.775i)T \)
13 \( 1 + (3.59 + 0.250i)T \)
good7 \( 1 + 0.946T + 7T^{2} \)
11 \( 1 + (-3.13 + 3.13i)T - 11iT^{2} \)
17 \( 1 + (-3.42 + 3.42i)T - 17iT^{2} \)
19 \( 1 + (-1.00 + 1.00i)T - 19iT^{2} \)
23 \( 1 + (4.25 + 4.25i)T + 23iT^{2} \)
29 \( 1 + 5.39iT - 29T^{2} \)
31 \( 1 + (2.43 + 2.43i)T + 31iT^{2} \)
37 \( 1 + 11.9T + 37T^{2} \)
41 \( 1 + (-6.03 - 6.03i)T + 41iT^{2} \)
43 \( 1 + (0.242 + 0.242i)T + 43iT^{2} \)
47 \( 1 - 0.854T + 47T^{2} \)
53 \( 1 + (-3.90 + 3.90i)T - 53iT^{2} \)
59 \( 1 + (9.38 + 9.38i)T + 59iT^{2} \)
61 \( 1 - 9.79T + 61T^{2} \)
67 \( 1 + 3.51iT - 67T^{2} \)
71 \( 1 + (6.99 + 6.99i)T + 71iT^{2} \)
73 \( 1 - 16.4iT - 73T^{2} \)
79 \( 1 + 8.09iT - 79T^{2} \)
83 \( 1 + 15.2T + 83T^{2} \)
89 \( 1 + (-0.244 - 0.244i)T + 89iT^{2} \)
97 \( 1 + 2.67iT - 97T^{2} \)
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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

−11.38366173926468118670476975550, −10.07457560267609964560415628693, −9.289003815751121623574507073168, −8.336662393144110401225879313031, −7.40407712824225577846165043284, −6.42003488492245504428556472704, −5.34635035551657169372074176396, −4.34865586833790299799298661098, −3.32424890271512670759848205969, −0.41222591774919527620918956090, 1.67823672526649456327706386586, 3.33088769557465018672302078716, 4.29233759497532087483794310520, 5.57576490756743095595202986461, 6.96876487449120939683903517694, 7.55412388215055330926317443537, 8.792592914695554355274595415839, 9.895014686857276189716621896982, 10.58289265932914463275444954448, 11.69814972018984561561907894049

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