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Properties

Label 2-760-760.659-c1-0-59
Degree $2$
Conductor $760$
Sign $-0.113 + 0.993i$
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  + (−0.909 − 1.08i)2-s + (−0.347 + 1.96i)4-s + (0.388 + 2.20i)5-s + (1.78 − 3.08i)7-s + (2.44 − 1.41i)8-s + (−2.29 − 1.92i)9-s + (2.03 − 2.42i)10-s + (−2.79 − 4.84i)11-s + (2.29 + 6.29i)13-s + (−4.96 + 0.874i)14-s + (−3.75 − 1.36i)16-s + 4.24i·18-s + (4.33 + 0.494i)19-s − 4.47·20-s + (−2.70 + 7.43i)22-s + (1.23 − 6.99i)23-s + ⋯
L(s)  = 1  + (−0.642 − 0.766i)2-s + (−0.173 + 0.984i)4-s + (0.173 + 0.984i)5-s + (0.673 − 1.16i)7-s + (0.866 − 0.500i)8-s + (−0.766 − 0.642i)9-s + (0.642 − 0.766i)10-s + (−0.843 − 1.46i)11-s + (0.635 + 1.74i)13-s + (−1.32 + 0.233i)14-s + (−0.939 − 0.342i)16-s + 0.999i·18-s + (0.993 + 0.113i)19-s − 0.999·20-s + (−0.577 + 1.58i)22-s + (0.257 − 1.45i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.113 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.113 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $-0.113 + 0.993i$
Analytic conductor: \(6.06863\)
Root analytic conductor: \(2.46345\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{760} (659, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 760,\ (\ :1/2),\ -0.113 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.671593 - 0.753063i\)
\(L(\frac12)\) \(\approx\) \(0.671593 - 0.753063i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.909 + 1.08i)T \)
5 \( 1 + (-0.388 - 2.20i)T \)
19 \( 1 + (-4.33 - 0.494i)T \)
good3 \( 1 + (2.29 + 1.92i)T^{2} \)
7 \( 1 + (-1.78 + 3.08i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.79 + 4.84i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.29 - 6.29i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-2.95 + 16.7i)T^{2} \)
23 \( 1 + (-1.23 + 6.99i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 12.1iT - 37T^{2} \)
41 \( 1 + (-3.75 + 10.3i)T + (-31.4 - 26.3i)T^{2} \)
43 \( 1 + (40.4 - 14.7i)T^{2} \)
47 \( 1 + (-3.26 - 2.73i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (-12.0 - 2.12i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (5.76 + 6.86i)T + (-10.2 + 58.1i)T^{2} \)
61 \( 1 + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-55.9 - 46.9i)T^{2} \)
79 \( 1 + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.42 + 6.66i)T + (-68.1 + 57.2i)T^{2} \)
97 \( 1 + (16.8 - 95.5i)T^{2} \)
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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

−10.42488403130856756148505137032, −9.217296503131872082545073266042, −8.603940144425325867333282077170, −7.59015914199178224754178049699, −6.86678414220252837854218805241, −5.78550037653923271405164544788, −4.15892049871446081317364712503, −3.42413139449148662964160521052, −2.31186611927435139928522547089, −0.69350796001639810273579754387, 1.38948167683195199789938913870, 2.66491015117430734220866840721, 4.87646338431729115866497024553, 5.31212377104086581334989892671, 5.84481644164940071571454474274, 7.53203263912690976611775046919, 8.048378868844640526030027427098, 8.634453742494030640289426891215, 9.585264172344208605987648615859, 10.24981997654312333629308172723

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