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Properties

Label 2-760-19.7-c1-0-8
Degree $2$
Conductor $760$
Sign $0.910 - 0.412i$
Analytic cond. $6.06863$
Root an. cond. $2.46345$
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)3-s + (−0.5 − 0.866i)5-s − 2·7-s + (1 − 1.73i)9-s + 5·11-s + (−3 + 5.19i)13-s + (0.499 − 0.866i)15-s + (3 + 5.19i)17-s + (3.5 − 2.59i)19-s + (−1 − 1.73i)21-s + (2 − 3.46i)23-s + (−0.499 + 0.866i)25-s + 5·27-s + (3 − 5.19i)29-s + 4·31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.223 − 0.387i)5-s − 0.755·7-s + (0.333 − 0.577i)9-s + 1.50·11-s + (−0.832 + 1.44i)13-s + (0.129 − 0.223i)15-s + (0.727 + 1.26i)17-s + (0.802 − 0.596i)19-s + (−0.218 − 0.377i)21-s + (0.417 − 0.722i)23-s + (−0.0999 + 0.173i)25-s + 0.962·27-s + (0.557 − 0.964i)29-s + 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $0.910 - 0.412i$
Analytic conductor: \(6.06863\)
Root analytic conductor: \(2.46345\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{760} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 760,\ (\ :1/2),\ 0.910 - 0.412i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63662 + 0.353718i\)
\(L(\frac12)\) \(\approx\) \(1.63662 + 0.353718i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-3.5 + 2.59i)T \)
good3 \( 1 + (-0.5 - 0.866i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + 2T + 7T^{2} \)
11 \( 1 - 5T + 11T^{2} \)
13 \( 1 + (3 - 5.19i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + (-5.5 - 9.52i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1 + 1.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.5 + 4.33i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.5 + 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (5 + 8.66i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.5 + 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3T + 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.5 + 7.79i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.03528996387769838848724582964, −9.453726057873517338723319940303, −9.050567273203785145574387973634, −7.899415781529398187808716609352, −6.65872552951647901265848798970, −6.29462411664765246763965647886, −4.57666272386209862509798759131, −4.08678819460603297512671945345, −2.99080406517359437205197455634, −1.23761206828327052267240077639, 1.07976426416212227132010067644, 2.75606273452331965250754120076, 3.48794920523652399613094655618, 4.91308000155256845909209041227, 5.92841901195640643908372510732, 7.18944077258408464064528057980, 7.34984261628592903619931828026, 8.484165536161377356117678801729, 9.633422391729570274715476782586, 10.03776481763495267659364773210

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