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Properties

Label 2-390-13.3-c1-0-2
Degree $2$
Conductor $390$
Sign $0.794 - 0.607i$
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  + (−0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + 5-s + (−0.499 + 0.866i)6-s + (−1.78 + 3.08i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (2.06 + 3.57i)11-s + 0.999·12-s + (−3.34 + 1.35i)13-s + 3.56·14-s + (−0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−2.56 + 4.43i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + 0.447·5-s + (−0.204 + 0.353i)6-s + (−0.673 + 1.16i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.158 − 0.273i)10-s + (0.621 + 1.07i)11-s + 0.288·12-s + (−0.926 + 0.375i)13-s + 0.951·14-s + (−0.129 − 0.223i)15-s + (−0.125 − 0.216i)16-s + (−0.621 + 1.07i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 - 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.794 - 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.794 - 0.607i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.794 - 0.607i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.790860 + 0.267662i\)
\(L(\frac12)\) \(\approx\) \(0.790860 + 0.267662i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 - T \)
13 \( 1 + (3.34 - 1.35i)T \)
good7 \( 1 + (1.78 - 3.08i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.06 - 3.57i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.56 - 4.43i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.78 + 3.08i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.84 - 6.65i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.28 + 5.68i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.68T + 31T^{2} \)
37 \( 1 + (2.06 + 3.57i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.12 - 3.67i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.28 - 3.95i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 7T + 47T^{2} \)
53 \( 1 - 4.43T + 53T^{2} \)
59 \( 1 + (5.28 - 9.14i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3 - 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.12 + 12.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.43 + 4.22i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 15.3T + 73T^{2} \)
79 \( 1 - 7.43T + 79T^{2} \)
83 \( 1 - 1.12T + 83T^{2} \)
89 \( 1 + (0.903 + 1.56i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.561 - 0.972i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.68867501396297936696666985870, −10.45440406621029518369843711679, −9.339057770244642815750541905145, −9.175009510060011017704772890301, −7.66115662753250876686988131040, −6.71074590948572232831897638643, −5.71596949922371053169128149454, −4.46075643872445643519425381494, −2.76576603052525779819467519888, −1.78480068823824731481418273688, 0.65060165622980218014313064976, 3.11629016998815189811666501121, 4.43809435009555833113135851153, 5.50569328611248731942388239493, 6.60617696497694105883191654303, 7.22766625826689786359455255311, 8.606061029348172105142517892364, 9.424156759263510281306151912106, 10.25376676553916992380158719646, 10.83975632795611182193630760698

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