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Properties

Label 2-390-13.12-c1-0-5
Degree $2$
Conductor $390$
Sign $0.987 - 0.155i$
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 + 3.12i·7-s + i·8-s + 9-s + 10-s + 5.12i·11-s − 12-s + (3.56 − 0.561i)13-s + 3.12·14-s + i·15-s + 16-s + 2·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577·3-s − 0.5·4-s + 0.447i·5-s − 0.408i·6-s + 1.18i·7-s + 0.353i·8-s + 0.333·9-s + 0.316·10-s + 1.54i·11-s − 0.288·12-s + (0.987 − 0.155i)13-s + 0.834·14-s + 0.258i·15-s + 0.250·16-s + 0.485·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.987 - 0.155i$
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: \((2,\ 390,\ (\ :1/2),\ 0.987 - 0.155i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56736 + 0.122805i\)
\(L(\frac12)\) \(\approx\) \(1.56736 + 0.122805i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.56 + 0.561i)T \)
good7 \( 1 - 3.12iT - 7T^{2} \)
11 \( 1 - 5.12iT - 11T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 6iT - 19T^{2} \)
23 \( 1 + 5.12T + 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 - 3.12iT - 31T^{2} \)
37 \( 1 + 5.12iT - 37T^{2} \)
41 \( 1 - 0.876iT - 41T^{2} \)
43 \( 1 - 6.24T + 43T^{2} \)
47 \( 1 + 6.24iT - 47T^{2} \)
53 \( 1 + 13.3T + 53T^{2} \)
59 \( 1 + 1.12iT - 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 - 4.87iT - 67T^{2} \)
71 \( 1 - 10.2iT - 71T^{2} \)
73 \( 1 + 13.1iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 6.24iT - 83T^{2} \)
89 \( 1 - 3.12iT - 89T^{2} \)
97 \( 1 + 13.1iT - 97T^{2} \)
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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

−11.38127746873682666631590746499, −10.37101883808793684106382668017, −9.519613973867561841431976654146, −8.812024689168253307926174133572, −7.80969928564689767813418066144, −6.64359584280747701279743440924, −5.36126612194867428113126943173, −4.14944062459346395365861817459, −2.87150561423351129406409523583, −1.94216972204415676821100824926, 1.12154252455089545342960236453, 3.47828406412126673822702117232, 4.17139879793489547440662405322, 5.70088274447625252525906654435, 6.49499788494832839476608606612, 8.022360486362146270906906186783, 8.072100298636340252821930282454, 9.297585462794563626732895733261, 10.24661716179834426251987970577, 11.13981131024825043284159099711

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