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Properties

Label 2-390-195.77-c1-0-4
Degree $2$
Conductor $390$
Sign $-0.745 - 0.666i$
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 + (0.805 + 1.53i)3-s + 1.00i·4-s + (−1.53 + 1.62i)5-s + (0.514 − 1.65i)6-s + (−0.677 + 0.677i)7-s + (0.707 − 0.707i)8-s + (−1.70 + 2.47i)9-s + (2.23 − 0.0614i)10-s − 2.37·11-s + (−1.53 + 0.805i)12-s + (0.799 − 3.51i)13-s + 0.957·14-s + (−3.72 − 1.04i)15-s − 1.00·16-s + (−5.31 + 5.31i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.465 + 0.885i)3-s + 0.500i·4-s + (−0.687 + 0.726i)5-s + (0.209 − 0.675i)6-s + (−0.255 + 0.255i)7-s + (0.250 − 0.250i)8-s + (−0.567 + 0.823i)9-s + (0.706 − 0.0194i)10-s − 0.716·11-s + (−0.442 + 0.232i)12-s + (0.221 − 0.975i)13-s + 0.255·14-s + (−0.962 − 0.270i)15-s − 0.250·16-s + (−1.29 + 1.29i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.745 - 0.666i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.745 - 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.745 - 0.666i$
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: \((2,\ 390,\ (\ :1/2),\ -0.745 - 0.666i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.228068 + 0.597402i\)
\(L(\frac12)\) \(\approx\) \(0.228068 + 0.597402i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.805 - 1.53i)T \)
5 \( 1 + (1.53 - 1.62i)T \)
13 \( 1 + (-0.799 + 3.51i)T \)
good7 \( 1 + (0.677 - 0.677i)T - 7iT^{2} \)
11 \( 1 + 2.37T + 11T^{2} \)
17 \( 1 + (5.31 - 5.31i)T - 17iT^{2} \)
19 \( 1 - 2.08T + 19T^{2} \)
23 \( 1 + (1.90 + 1.90i)T + 23iT^{2} \)
29 \( 1 + 1.89T + 29T^{2} \)
31 \( 1 - 8.49iT - 31T^{2} \)
37 \( 1 + (2.64 - 2.64i)T - 37iT^{2} \)
41 \( 1 - 8.75T + 41T^{2} \)
43 \( 1 + (-0.132 + 0.132i)T - 43iT^{2} \)
47 \( 1 + (2.08 + 2.08i)T + 47iT^{2} \)
53 \( 1 + (-5.82 - 5.82i)T + 53iT^{2} \)
59 \( 1 - 11.2iT - 59T^{2} \)
61 \( 1 - 9.29T + 61T^{2} \)
67 \( 1 + (8.47 - 8.47i)T - 67iT^{2} \)
71 \( 1 - 12.4T + 71T^{2} \)
73 \( 1 + (3.69 + 3.69i)T + 73iT^{2} \)
79 \( 1 + 1.03iT - 79T^{2} \)
83 \( 1 + (-0.475 + 0.475i)T - 83iT^{2} \)
89 \( 1 + 8.15iT - 89T^{2} \)
97 \( 1 + (-1.11 + 1.11i)T - 97iT^{2} \)
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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

−11.26019928605449140579334441873, −10.57569206207869880846906820915, −10.16454628445911665161496099522, −8.838538610881353255898367947115, −8.271475872488528289120442100046, −7.30338536484092009325357966240, −5.87788469956673588082453578355, −4.42250534701184834396399272374, −3.38932511292656623648251354259, −2.50864943199648952861637208852, 0.44938352279062316870289884666, 2.20523333190926941008369924849, 3.92348261663006997493660561123, 5.23547465956639080666079560152, 6.54815496105941759166985286201, 7.38169410034767460401411756644, 8.037154578671862019932798463497, 9.048479671694441996409717352213, 9.557748688778262202674494540392, 11.21932856432670823774032661098

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