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Properties

Label 2-760-760.299-c1-0-48
Degree $2$
Conductor $760$
Sign $0.726 - 0.687i$
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  + (0.483 + 1.32i)2-s + (−1.53 + 1.28i)4-s + (−1.71 − 1.43i)5-s + (−0.803 − 1.39i)7-s + (−2.44 − 1.41i)8-s + (2.81 + 1.02i)9-s + (1.08 − 2.97i)10-s + (−0.141 + 0.245i)11-s + (6.59 − 1.16i)13-s + (1.46 − 1.74i)14-s + (0.694 − 3.93i)16-s + 4.24i·18-s + (−1.73 + 3.99i)19-s + 4.47·20-s + (−0.395 − 0.0696i)22-s + (6.99 − 5.87i)23-s + ⋯
L(s)  = 1  + (0.342 + 0.939i)2-s + (−0.766 + 0.642i)4-s + (−0.766 − 0.642i)5-s + (−0.303 − 0.526i)7-s + (−0.866 − 0.499i)8-s + (0.939 + 0.342i)9-s + (0.342 − 0.939i)10-s + (−0.0427 + 0.0740i)11-s + (1.83 − 0.322i)13-s + (0.390 − 0.465i)14-s + (0.173 − 0.984i)16-s + 0.999i·18-s + (−0.398 + 0.917i)19-s + 0.999·20-s + (−0.0842 − 0.0148i)22-s + (1.45 − 1.22i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 - 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.726 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $0.726 - 0.687i$
Analytic conductor: \(6.06863\)
Root analytic conductor: \(2.46345\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{760} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 760,\ (\ :1/2),\ 0.726 - 0.687i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45799 + 0.580315i\)
\(L(\frac12)\) \(\approx\) \(1.45799 + 0.580315i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.483 - 1.32i)T \)
5 \( 1 + (1.71 + 1.43i)T \)
19 \( 1 + (1.73 - 3.99i)T \)
good3 \( 1 + (-2.81 - 1.02i)T^{2} \)
7 \( 1 + (0.803 + 1.39i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.141 - 0.245i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-6.59 + 1.16i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (-13.0 + 10.9i)T^{2} \)
23 \( 1 + (-6.99 + 5.87i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.36iT - 37T^{2} \)
41 \( 1 + (-11.0 - 1.94i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (-7.46 - 42.3i)T^{2} \)
47 \( 1 + (-4.00 - 1.45i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (8.32 + 9.92i)T + (-9.20 + 52.1i)T^{2} \)
59 \( 1 + (3.06 + 8.42i)T + (-45.1 + 37.9i)T^{2} \)
61 \( 1 + (-10.5 + 60.0i)T^{2} \)
67 \( 1 + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (68.5 + 24.9i)T^{2} \)
79 \( 1 + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-11.4 + 2.01i)T + (83.6 - 30.4i)T^{2} \)
97 \( 1 + (74.3 - 62.3i)T^{2} \)
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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

−10.43828313849610349248737765590, −9.316015419900128467947355363071, −8.425206466317981941935710515271, −7.900797360634070866026336289424, −6.92966215164902976981995984495, −6.16362034058409195750525762377, −4.93925813117833707293392418724, −4.16412139351880693690530265928, −3.41320871851091969549473722296, −0.991701139643589788593989769434, 1.14326961884991320554332963846, 2.72967088451281798691828556506, 3.67804219705070966618360670685, 4.37327174410965545842839323957, 5.75026588903763773615460271583, 6.60940702423581236708337269980, 7.64105814580367118930212021232, 9.019900349670795409652165704729, 9.222173123670008143857397946937, 10.72756703159094185004240794351

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