Properties

Label 2-390-39.8-c1-0-14
Degree $2$
Conductor $390$
Sign $-0.614 + 0.789i$
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  + (0.707 − 0.707i)2-s + (0.292 − 1.70i)3-s − 1.00i·4-s + (−0.707 + 0.707i)5-s + (−0.999 − 1.41i)6-s + (2 − 2i)7-s + (−0.707 − 0.707i)8-s + (−2.82 − i)9-s + 1.00i·10-s + (−2 − 2i)11-s + (−1.70 − 0.292i)12-s + (3.53 + 0.707i)13-s − 2.82i·14-s + (0.999 + 1.41i)15-s − 1.00·16-s − 4.82·17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.169 − 0.985i)3-s − 0.500i·4-s + (−0.316 + 0.316i)5-s + (−0.408 − 0.577i)6-s + (0.755 − 0.755i)7-s + (−0.250 − 0.250i)8-s + (−0.942 − 0.333i)9-s + 0.316i·10-s + (−0.603 − 0.603i)11-s + (−0.492 − 0.0845i)12-s + (0.980 + 0.196i)13-s − 0.755i·14-s + (0.258 + 0.365i)15-s − 0.250·16-s − 1.17·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.614 + 0.789i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.614 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.614 + 0.789i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (281, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ -0.614 + 0.789i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.764176 - 1.56300i\)
\(L(\frac12)\) \(\approx\) \(0.764176 - 1.56300i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (-0.292 + 1.70i)T \)
5 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 + (-3.53 - 0.707i)T \)
good7 \( 1 + (-2 + 2i)T - 7iT^{2} \)
11 \( 1 + (2 + 2i)T + 11iT^{2} \)
17 \( 1 + 4.82T + 17T^{2} \)
19 \( 1 + 19iT^{2} \)
23 \( 1 - 6.82T + 23T^{2} \)
29 \( 1 + 3.65iT - 29T^{2} \)
31 \( 1 + (-4.41 - 4.41i)T + 31iT^{2} \)
37 \( 1 + (4 - 4i)T - 37iT^{2} \)
41 \( 1 + (-2.17 + 2.17i)T - 41iT^{2} \)
43 \( 1 + 5.07iT - 43T^{2} \)
47 \( 1 + (-2 - 2i)T + 47iT^{2} \)
53 \( 1 - 9.41iT - 53T^{2} \)
59 \( 1 + (-4 - 4i)T + 59iT^{2} \)
61 \( 1 - 12.8T + 61T^{2} \)
67 \( 1 + (-2.24 - 2.24i)T + 67iT^{2} \)
71 \( 1 + (-10.4 + 10.4i)T - 71iT^{2} \)
73 \( 1 + (-9.89 + 9.89i)T - 73iT^{2} \)
79 \( 1 + 8.82T + 79T^{2} \)
83 \( 1 + (1.17 - 1.17i)T - 83iT^{2} \)
89 \( 1 + (-2.65 - 2.65i)T + 89iT^{2} \)
97 \( 1 + (8.58 + 8.58i)T + 97iT^{2} \)
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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

−11.04692834325106249845418495439, −10.64549631681346216845201585761, −8.942216359246531956020088592940, −8.185400764787085542435244941696, −7.13197788453849232604205647059, −6.31214206160717987261306587479, −5.02840338038339822089732637621, −3.75804137270984099687256149983, −2.55084595636004700514206803790, −1.05054651183583015538093518715, 2.49752041749616048376467577338, 3.88605592448865906594288773242, 4.89159059447817598370733025184, 5.47834947827813058769572161186, 6.85225655755882093286751277852, 8.264786965042379272446274953611, 8.606076713292528516903228581338, 9.672101062335020052427327239693, 11.03847047404108062280958309617, 11.37924482079677062240585348981

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