Properties

Label 2-760-8.5-c1-0-69
Degree $2$
Conductor $760$
Sign $-0.878 - 0.477i$
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  + (0.899 − 1.09i)2-s − 3.28i·3-s + (−0.381 − 1.96i)4-s − i·5-s + (−3.58 − 2.95i)6-s + 3.22·7-s + (−2.48 − 1.34i)8-s − 7.78·9-s + (−1.09 − 0.899i)10-s + 2.17i·11-s + (−6.44 + 1.25i)12-s − 0.674i·13-s + (2.90 − 3.51i)14-s − 3.28·15-s + (−3.70 + 1.49i)16-s + 6.83·17-s + ⋯
L(s)  = 1  + (0.636 − 0.771i)2-s − 1.89i·3-s + (−0.190 − 0.981i)4-s − 0.447i·5-s + (−1.46 − 1.20i)6-s + 1.21·7-s + (−0.878 − 0.477i)8-s − 2.59·9-s + (−0.345 − 0.284i)10-s + 0.655i·11-s + (−1.86 + 0.361i)12-s − 0.187i·13-s + (0.775 − 0.940i)14-s − 0.848·15-s + (−0.927 + 0.374i)16-s + 1.65·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.878 - 0.477i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.878 - 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $-0.878 - 0.477i$
Analytic conductor: \(6.06863\)
Root analytic conductor: \(2.46345\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{760} (381, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 760,\ (\ :1/2),\ -0.878 - 0.477i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.528365 + 2.08044i\)
\(L(\frac12)\) \(\approx\) \(0.528365 + 2.08044i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.899 + 1.09i)T \)
5 \( 1 + iT \)
19 \( 1 + iT \)
good3 \( 1 + 3.28iT - 3T^{2} \)
7 \( 1 - 3.22T + 7T^{2} \)
11 \( 1 - 2.17iT - 11T^{2} \)
13 \( 1 + 0.674iT - 13T^{2} \)
17 \( 1 - 6.83T + 17T^{2} \)
23 \( 1 + 4.67T + 23T^{2} \)
29 \( 1 + 1.53iT - 29T^{2} \)
31 \( 1 - 2.06T + 31T^{2} \)
37 \( 1 + 5.49iT - 37T^{2} \)
41 \( 1 - 10.7T + 41T^{2} \)
43 \( 1 - 11.3iT - 43T^{2} \)
47 \( 1 - 11.2T + 47T^{2} \)
53 \( 1 - 1.67iT - 53T^{2} \)
59 \( 1 + 13.0iT - 59T^{2} \)
61 \( 1 - 4.92iT - 61T^{2} \)
67 \( 1 - 5.07iT - 67T^{2} \)
71 \( 1 + 13.6T + 71T^{2} \)
73 \( 1 + 5.36T + 73T^{2} \)
79 \( 1 + 11.1T + 79T^{2} \)
83 \( 1 + 4.68iT - 83T^{2} \)
89 \( 1 + 4.65T + 89T^{2} \)
97 \( 1 - 1.56T + 97T^{2} \)
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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

−9.994539541689584550049829649716, −8.855287408615588067768117820134, −7.88234165783149322729410633701, −7.41443368393855358753969019512, −6.05086471534773529965376023580, −5.50397212845484869770937818435, −4.35279151405180952006724090941, −2.76529788217432713595087056225, −1.77350619800009222431270079341, −0.965923631416648630764003130696, 2.82589657392771601959783781668, 3.79446318084179745897847734641, 4.46971744584274466847583780860, 5.50251172597178900653074348789, 5.88119204780895331327368799767, 7.52991142457764457157096165974, 8.295286588105585864232161222834, 9.000008246668808331085710863029, 10.06480203470155490319358905133, 10.74202426707516305561133915597

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