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Properties

Label 2-390-15.8-c1-0-2
Degree $2$
Conductor $390$
Sign $-0.374 - 0.927i$
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 + (−1.41 + i)3-s + 1.00i·4-s + (0.707 + 2.12i)5-s + (1.70 + 0.292i)6-s + (0.585 − 0.585i)7-s + (0.707 − 0.707i)8-s + (1.00 − 2.82i)9-s + (0.999 − 2i)10-s + 4i·11-s + (−1.00 − 1.41i)12-s + (−0.707 − 0.707i)13-s − 0.828·14-s + (−3.12 − 2.29i)15-s − 1.00·16-s + (−3 − 3i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.816 + 0.577i)3-s + 0.500i·4-s + (0.316 + 0.948i)5-s + (0.696 + 0.119i)6-s + (0.221 − 0.221i)7-s + (0.250 − 0.250i)8-s + (0.333 − 0.942i)9-s + (0.316 − 0.632i)10-s + 1.20i·11-s + (−0.288 − 0.408i)12-s + (−0.196 − 0.196i)13-s − 0.221·14-s + (−0.805 − 0.592i)15-s − 0.250·16-s + (−0.727 − 0.727i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.374 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.374 - 0.927i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ -0.374 - 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.345558 + 0.512136i\)
\(L(\frac12)\) \(\approx\) \(0.345558 + 0.512136i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.41 - i)T \)
5 \( 1 + (-0.707 - 2.12i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + (-0.585 + 0.585i)T - 7iT^{2} \)
11 \( 1 - 4iT - 11T^{2} \)
17 \( 1 + (3 + 3i)T + 17iT^{2} \)
19 \( 1 - 6.82iT - 19T^{2} \)
23 \( 1 + (2.82 - 2.82i)T - 23iT^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 3.41T + 31T^{2} \)
37 \( 1 + (2.58 - 2.58i)T - 37iT^{2} \)
41 \( 1 - 2iT - 41T^{2} \)
43 \( 1 + (-1.24 - 1.24i)T + 43iT^{2} \)
47 \( 1 + (2.24 + 2.24i)T + 47iT^{2} \)
53 \( 1 + (-0.585 + 0.585i)T - 53iT^{2} \)
59 \( 1 - 4.48T + 59T^{2} \)
61 \( 1 - 4.82T + 61T^{2} \)
67 \( 1 + (0.828 - 0.828i)T - 67iT^{2} \)
71 \( 1 + 14.2iT - 71T^{2} \)
73 \( 1 + (11.0 + 11.0i)T + 73iT^{2} \)
79 \( 1 - 12.4iT - 79T^{2} \)
83 \( 1 + (6.82 - 6.82i)T - 83iT^{2} \)
89 \( 1 - 11.1T + 89T^{2} \)
97 \( 1 + (-9.07 + 9.07i)T - 97iT^{2} \)
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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

−11.47972028688588445839458255607, −10.53357092643256446378856799892, −9.989481760553002633866796684891, −9.345914280432447398023532416183, −7.77066286457217796851671203475, −6.95647628325603408388792763229, −5.88377863499871455464364409861, −4.60588035208488916483924363412, −3.49414023136330298830874490160, −1.91208173192140746515036717592, 0.52324226545872310183666597743, 2.05893188539416230483846160679, 4.46676605468696305797283336495, 5.47730516189928737245170893962, 6.19074691674378051298019067888, 7.20648646356412786371071650270, 8.420480113526148456190858538482, 8.859468425361315376454223034291, 10.12425617522756951133904276913, 11.12741504192515361248153307657

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