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Properties

Label 2-390-195.77-c1-0-5
Degree $2$
Conductor $390$
Sign $-0.540 - 0.841i$
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 + (−0.618 − 1.61i)3-s + 1.00i·4-s + (0.707 + 2.12i)5-s + (0.707 − 1.58i)6-s + (−3.16 + 3.16i)7-s + (−0.707 + 0.707i)8-s + (−2.23 + 2.00i)9-s + (−0.999 + 2i)10-s − 5.65·11-s + (1.61 − 0.618i)12-s + (3.58 − 0.418i)13-s − 4.47·14-s + (2.99 − 2.45i)15-s − 1.00·16-s + (−2.23 + 2.23i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.356 − 0.934i)3-s + 0.500i·4-s + (0.316 + 0.948i)5-s + (0.288 − 0.645i)6-s + (−1.19 + 1.19i)7-s + (−0.250 + 0.250i)8-s + (−0.745 + 0.666i)9-s + (−0.316 + 0.632i)10-s − 1.70·11-s + (0.467 − 0.178i)12-s + (0.993 − 0.116i)13-s − 1.19·14-s + (0.773 − 0.633i)15-s − 0.250·16-s + (−0.542 + 0.542i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.540 - 0.841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.540 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.540 - 0.841i$
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: \((2,\ 390,\ (\ :1/2),\ -0.540 - 0.841i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.510007 + 0.934037i\)
\(L(\frac12)\) \(\approx\) \(0.510007 + 0.934037i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.618 + 1.61i)T \)
5 \( 1 + (-0.707 - 2.12i)T \)
13 \( 1 + (-3.58 + 0.418i)T \)
good7 \( 1 + (3.16 - 3.16i)T - 7iT^{2} \)
11 \( 1 + 5.65T + 11T^{2} \)
17 \( 1 + (2.23 - 2.23i)T - 17iT^{2} \)
19 \( 1 - 6.32T + 19T^{2} \)
23 \( 1 + 23iT^{2} \)
29 \( 1 - 4.47T + 29T^{2} \)
31 \( 1 + 3.16iT - 31T^{2} \)
37 \( 1 + (3.16 - 3.16i)T - 37iT^{2} \)
41 \( 1 - 5.65T + 41T^{2} \)
43 \( 1 + (5 - 5i)T - 43iT^{2} \)
47 \( 1 + (-1.41 - 1.41i)T + 47iT^{2} \)
53 \( 1 + (-4.47 - 4.47i)T + 53iT^{2} \)
59 \( 1 + 2.82iT - 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 - 67iT^{2} \)
71 \( 1 + 1.41T + 71T^{2} \)
73 \( 1 + (3.16 + 3.16i)T + 73iT^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + (2.82 - 2.82i)T - 83iT^{2} \)
89 \( 1 - 2.82iT - 89T^{2} \)
97 \( 1 + (9.48 - 9.48i)T - 97iT^{2} \)
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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

−11.79017789854462338594837366394, −10.89519772734850092841453188623, −9.870741538831767237048759062078, −8.581501612087711735231123333451, −7.67761866008252270667215249051, −6.66504212139714825026163778478, −5.96847056716911369310177850429, −5.36532752723958953232891571674, −3.17676103604724935161457509780, −2.48500885587855024999732704235, 0.60227342985266670886085264104, 2.99175151990071178374167966840, 3.98673539020263390472149272781, 5.03033984905537727886118687725, 5.77376900602139289872376939373, 7.03521857743817452817854205385, 8.526021835456742939904116686811, 9.568685542455019803730018184355, 10.18616073828428117442721543536, 10.81546199466636148004517148061

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