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Properties

Label 2-390-195.89-c1-0-22
Degree $2$
Conductor $390$
Sign $0.367 + 0.930i$
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  + (0.258 − 0.965i)2-s + (1.70 − 0.326i)3-s + (−0.866 − 0.499i)4-s + (2.00 − 0.995i)5-s + (0.125 − 1.72i)6-s + (0.991 − 0.265i)7-s + (−0.707 + 0.707i)8-s + (2.78 − 1.10i)9-s + (−0.443 − 2.19i)10-s + (−1.65 + 6.15i)11-s + (−1.63 − 0.568i)12-s + (3.21 − 1.63i)13-s − 1.02i·14-s + (3.08 − 2.34i)15-s + (0.500 + 0.866i)16-s + (−5.23 − 3.02i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (0.982 − 0.188i)3-s + (−0.433 − 0.249i)4-s + (0.895 − 0.445i)5-s + (0.0511 − 0.705i)6-s + (0.374 − 0.100i)7-s + (−0.249 + 0.249i)8-s + (0.929 − 0.369i)9-s + (−0.140 − 0.693i)10-s + (−0.497 + 1.85i)11-s + (−0.472 − 0.164i)12-s + (0.890 − 0.454i)13-s − 0.274i·14-s + (0.795 − 0.605i)15-s + (0.125 + 0.216i)16-s + (−1.26 − 0.732i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 + 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.367 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.367 + 0.930i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.367 + 0.930i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83636 - 1.24936i\)
\(L(\frac12)\) \(\approx\) \(1.83636 - 1.24936i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
3 \( 1 + (-1.70 + 0.326i)T \)
5 \( 1 + (-2.00 + 0.995i)T \)
13 \( 1 + (-3.21 + 1.63i)T \)
good7 \( 1 + (-0.991 + 0.265i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (1.65 - 6.15i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (5.23 + 3.02i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.34 - 1.69i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (1.26 - 0.728i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.41 - 2.54i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.46 - 1.46i)T + 31iT^{2} \)
37 \( 1 + (-1.76 + 6.57i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-2.67 - 0.717i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (1.80 - 3.13i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.16 + 5.16i)T - 47iT^{2} \)
53 \( 1 + 8.98T + 53T^{2} \)
59 \( 1 + (-3.52 + 0.943i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (4.69 - 8.12i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.04 + 2.15i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.360 - 1.34i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-8.65 - 8.65i)T + 73iT^{2} \)
79 \( 1 - 6.23T + 79T^{2} \)
83 \( 1 + (-5.68 - 5.68i)T + 83iT^{2} \)
89 \( 1 + (-2.95 + 11.0i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-1.09 - 4.10i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.95019046711579734349503740518, −10.16664931141149103966857438684, −9.342855804216607603650653071175, −8.649395881354740846181500098030, −7.56467815620799433508965463932, −6.38388085319775645917042297432, −4.91718569401558101548509093872, −4.11893029690092073543851182399, −2.43575574466493050935552719158, −1.73275675457414174100411324974, 2.10087236619864478459828796984, 3.39480049296547300757264247070, 4.55817202009794314806506799817, 6.01631901136381505511455386219, 6.53338694108585838943490415793, 8.041707679816667948049156745653, 8.612085011671996706635649675513, 9.304071563158327793904576001304, 10.66206276670407625096858373549, 11.10004504681693003940049436584

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