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Properties

Label 2-390-65.18-c1-0-3
Degree $2$
Conductor $390$
Sign $-0.733 - 0.679i$
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 + (−0.623 + 2.14i)5-s + (−0.707 + 0.707i)6-s + 1.64·7-s − i·8-s + 1.00i·9-s + (−2.14 − 0.623i)10-s + (1.53 + 1.53i)11-s + (−0.707 − 0.707i)12-s + (−3.01 + 1.97i)13-s + 1.64i·14-s + (−1.95 + 1.07i)15-s + 16-s + (2.27 + 2.27i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.408 + 0.408i)3-s − 0.5·4-s + (−0.279 + 0.960i)5-s + (−0.288 + 0.288i)6-s + 0.620·7-s − 0.353i·8-s + 0.333i·9-s + (−0.679 − 0.197i)10-s + (0.461 + 0.461i)11-s + (−0.204 − 0.204i)12-s + (−0.836 + 0.548i)13-s + 0.438i·14-s + (−0.505 + 0.278i)15-s + 0.250·16-s + (0.552 + 0.552i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.733 - 0.679i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.733 - 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.733 - 0.679i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ -0.733 - 0.679i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.505512 + 1.28970i\)
\(L(\frac12)\) \(\approx\) \(0.505512 + 1.28970i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.623 - 2.14i)T \)
13 \( 1 + (3.01 - 1.97i)T \)
good7 \( 1 - 1.64T + 7T^{2} \)
11 \( 1 + (-1.53 - 1.53i)T + 11iT^{2} \)
17 \( 1 + (-2.27 - 2.27i)T + 17iT^{2} \)
19 \( 1 + (3.55 + 3.55i)T + 19iT^{2} \)
23 \( 1 + (2.02 - 2.02i)T - 23iT^{2} \)
29 \( 1 - 2.17iT - 29T^{2} \)
31 \( 1 + (-1 + i)T - 31iT^{2} \)
37 \( 1 - 7.18T + 37T^{2} \)
41 \( 1 + (-0.918 + 0.918i)T - 41iT^{2} \)
43 \( 1 + (-3.99 + 3.99i)T - 43iT^{2} \)
47 \( 1 - 12.3T + 47T^{2} \)
53 \( 1 + (-0.679 - 0.679i)T + 53iT^{2} \)
59 \( 1 + (-0.278 + 0.278i)T - 59iT^{2} \)
61 \( 1 - 2.98T + 61T^{2} \)
67 \( 1 - 8.58iT - 67T^{2} \)
71 \( 1 + (-2.16 + 2.16i)T - 71iT^{2} \)
73 \( 1 + 13.0iT - 73T^{2} \)
79 \( 1 - 7.32iT - 79T^{2} \)
83 \( 1 - 15.4T + 83T^{2} \)
89 \( 1 + (-3.16 + 3.16i)T - 89iT^{2} \)
97 \( 1 + 15.2iT - 97T^{2} \)
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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

−11.56300117515043355381197044369, −10.62752779724499310248666727930, −9.754001760861716275313957244802, −8.845579106140508579740235010559, −7.77117346884582213927927820406, −7.13614893550126040733895794490, −6.06182600332210267513147756499, −4.70283604372716098650121308689, −3.85391046608208062611102508985, −2.31500871691703349190654909656, 0.937071821559497292547772200639, 2.36966186154445499391598212084, 3.84838225719027711240493919156, 4.84523550245659323813635549227, 5.99471922890965399521238622105, 7.66314459346667156079426584193, 8.209141884327240214493578506296, 9.133230019265063570118189357775, 9.991037234433498539139558253432, 11.13371109287464992467282714746

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