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Properties

Label 2-390-195.77-c1-0-0
Degree $2$
Conductor $390$
Sign $-0.999 - 0.0430i$
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 + (−1.50 + 0.854i)3-s + 1.00i·4-s + (1.11 + 1.93i)5-s + (1.66 + 0.461i)6-s + (−3.10 + 3.10i)7-s + (0.707 − 0.707i)8-s + (1.54 − 2.57i)9-s + (0.582 − 2.15i)10-s − 0.330·11-s + (−0.854 − 1.50i)12-s + (−2.38 − 2.70i)13-s + 4.39·14-s + (−3.33 − 1.96i)15-s − 1.00·16-s + (1.46 − 1.46i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.869 + 0.493i)3-s + 0.500i·4-s + (0.498 + 0.866i)5-s + (0.681 + 0.188i)6-s + (−1.17 + 1.17i)7-s + (0.250 − 0.250i)8-s + (0.513 − 0.858i)9-s + (0.184 − 0.682i)10-s − 0.0995·11-s + (−0.246 − 0.434i)12-s + (−0.660 − 0.750i)13-s + 1.17·14-s + (−0.861 − 0.508i)15-s − 0.250·16-s + (0.355 − 0.355i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0430i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.999 - 0.0430i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ -0.999 - 0.0430i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00427600 + 0.198738i\)
\(L(\frac12)\) \(\approx\) \(0.00427600 + 0.198738i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.50 - 0.854i)T \)
5 \( 1 + (-1.11 - 1.93i)T \)
13 \( 1 + (2.38 + 2.70i)T \)
good7 \( 1 + (3.10 - 3.10i)T - 7iT^{2} \)
11 \( 1 + 0.330T + 11T^{2} \)
17 \( 1 + (-1.46 + 1.46i)T - 17iT^{2} \)
19 \( 1 + 3.51T + 19T^{2} \)
23 \( 1 + (5.05 + 5.05i)T + 23iT^{2} \)
29 \( 1 - 4.75T + 29T^{2} \)
31 \( 1 + 1.27iT - 31T^{2} \)
37 \( 1 + (5.54 - 5.54i)T - 37iT^{2} \)
41 \( 1 + 9.24T + 41T^{2} \)
43 \( 1 + (3.66 - 3.66i)T - 43iT^{2} \)
47 \( 1 + (-5.05 - 5.05i)T + 47iT^{2} \)
53 \( 1 + (4.90 + 4.90i)T + 53iT^{2} \)
59 \( 1 + 6.37iT - 59T^{2} \)
61 \( 1 + 0.546T + 61T^{2} \)
67 \( 1 + (0.786 - 0.786i)T - 67iT^{2} \)
71 \( 1 - 0.813T + 71T^{2} \)
73 \( 1 + (-7.14 - 7.14i)T + 73iT^{2} \)
79 \( 1 - 13.7iT - 79T^{2} \)
83 \( 1 + (-6.61 + 6.61i)T - 83iT^{2} \)
89 \( 1 + 7.13iT - 89T^{2} \)
97 \( 1 + (-2.78 + 2.78i)T - 97iT^{2} \)
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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

−11.76235886067416755404714871504, −10.56378972390694382654216934721, −10.06361704501264122315124327160, −9.472430745859974040233317828063, −8.290082683104547211108245620424, −6.74491891050287814678006301493, −6.18822035950512763416564088479, −5.07573156563947709741319026460, −3.40635257678412391917834266351, −2.44719778697803953677064465294, 0.16408810971565859359832930090, 1.71905284374938237563020721422, 4.08954330570896061909252232620, 5.24878337650174258259260762477, 6.28015126360695400238593135964, 6.94447374518696913802290509759, 7.88823149964766466695720591910, 9.106271575097458021193857911486, 10.10687347167996147416795823660, 10.45234763494557788645151537848

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