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Properties

Label 2-390-195.77-c1-0-3
Degree $2$
Conductor $390$
Sign $0.632 - 0.774i$
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 + (−1.72 + 0.128i)3-s + 1.00i·4-s + (−1.56 − 1.60i)5-s + (1.31 + 1.13i)6-s + (−0.236 + 0.236i)7-s + (0.707 − 0.707i)8-s + (2.96 − 0.445i)9-s + (−0.0281 + 2.23i)10-s − 2.01·11-s + (−0.128 − 1.72i)12-s + (−3.47 + 0.965i)13-s + 0.334·14-s + (2.90 + 2.56i)15-s − 1.00·16-s + (−2.69 + 2.69i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.997 + 0.0743i)3-s + 0.500i·4-s + (−0.698 − 0.715i)5-s + (0.535 + 0.461i)6-s + (−0.0893 + 0.0893i)7-s + (0.250 − 0.250i)8-s + (0.988 − 0.148i)9-s + (−0.00888 + 0.707i)10-s − 0.606·11-s + (−0.0371 − 0.498i)12-s + (−0.963 + 0.267i)13-s + 0.0893·14-s + (0.749 + 0.662i)15-s − 0.250·16-s + (−0.652 + 0.652i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.632 - 0.774i$
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: \((2,\ 390,\ (\ :1/2),\ 0.632 - 0.774i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.387291 + 0.183896i\)
\(L(\frac12)\) \(\approx\) \(0.387291 + 0.183896i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.72 - 0.128i)T \)
5 \( 1 + (1.56 + 1.60i)T \)
13 \( 1 + (3.47 - 0.965i)T \)
good7 \( 1 + (0.236 - 0.236i)T - 7iT^{2} \)
11 \( 1 + 2.01T + 11T^{2} \)
17 \( 1 + (2.69 - 2.69i)T - 17iT^{2} \)
19 \( 1 - 8.01T + 19T^{2} \)
23 \( 1 + (-6.23 - 6.23i)T + 23iT^{2} \)
29 \( 1 - 4.55T + 29T^{2} \)
31 \( 1 - 7.05iT - 31T^{2} \)
37 \( 1 + (1.19 - 1.19i)T - 37iT^{2} \)
41 \( 1 + 7.90T + 41T^{2} \)
43 \( 1 + (-2.76 + 2.76i)T - 43iT^{2} \)
47 \( 1 + (-2.04 - 2.04i)T + 47iT^{2} \)
53 \( 1 + (6.83 + 6.83i)T + 53iT^{2} \)
59 \( 1 - 12.3iT - 59T^{2} \)
61 \( 1 - 5.48T + 61T^{2} \)
67 \( 1 + (-4.26 + 4.26i)T - 67iT^{2} \)
71 \( 1 + 9.88T + 71T^{2} \)
73 \( 1 + (2.59 + 2.59i)T + 73iT^{2} \)
79 \( 1 + 10.6iT - 79T^{2} \)
83 \( 1 + (2.98 - 2.98i)T - 83iT^{2} \)
89 \( 1 - 1.02iT - 89T^{2} \)
97 \( 1 + (10.9 - 10.9i)T - 97iT^{2} \)
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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

−11.53617643967983869854207270597, −10.62936968022657485360098535077, −9.713655342130819948317509160507, −8.882913140159821500923430551956, −7.65369074542363062434620855747, −6.98702537242194631615739438774, −5.35871686148053001061852769519, −4.69356658784969155525685996214, −3.27713576568980162281842916403, −1.26161394024179732240234973397, 0.42996288439946438894194342627, 2.79469435024722999005706065912, 4.58646602608554490367607387862, 5.42171577756277967293542765003, 6.77284670830589416201125683449, 7.20688765244502524165191892201, 8.126355950008814436937005592331, 9.576296341161241859407727066273, 10.29729967742973444514824843840, 11.15140796851323185245344480669

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