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Properties

Label 2-760-8.5-c1-0-8
Degree $2$
Conductor $760$
Sign $-0.302 - 0.953i$
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  + (−1.29 − 0.577i)2-s + 1.40i·3-s + (1.33 + 1.49i)4-s − i·5-s + (0.814 − 1.81i)6-s − 1.07·7-s + (−0.856 − 2.69i)8-s + 1.01·9-s + (−0.577 + 1.29i)10-s + 2.73i·11-s + (−2.10 + 1.87i)12-s + 2.56i·13-s + (1.39 + 0.622i)14-s + 1.40·15-s + (−0.451 + 3.97i)16-s − 7.57·17-s + ⋯
L(s)  = 1  + (−0.912 − 0.408i)2-s + 0.813i·3-s + (0.665 + 0.745i)4-s − 0.447i·5-s + (0.332 − 0.742i)6-s − 0.407·7-s + (−0.302 − 0.953i)8-s + 0.337·9-s + (−0.182 + 0.408i)10-s + 0.824i·11-s + (−0.607 + 0.542i)12-s + 0.710i·13-s + (0.371 + 0.166i)14-s + 0.364·15-s + (−0.112 + 0.993i)16-s − 1.83·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.302 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.302 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $-0.302 - 0.953i$
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: \((2,\ 760,\ (\ :1/2),\ -0.302 - 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.414507 + 0.566721i\)
\(L(\frac12)\) \(\approx\) \(0.414507 + 0.566721i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.29 + 0.577i)T \)
5 \( 1 + iT \)
19 \( 1 + iT \)
good3 \( 1 - 1.40iT - 3T^{2} \)
7 \( 1 + 1.07T + 7T^{2} \)
11 \( 1 - 2.73iT - 11T^{2} \)
13 \( 1 - 2.56iT - 13T^{2} \)
17 \( 1 + 7.57T + 17T^{2} \)
23 \( 1 - 5.41T + 23T^{2} \)
29 \( 1 - 3.77iT - 29T^{2} \)
31 \( 1 - 7.99T + 31T^{2} \)
37 \( 1 - 5.91iT - 37T^{2} \)
41 \( 1 + 8.09T + 41T^{2} \)
43 \( 1 - 2.43iT - 43T^{2} \)
47 \( 1 + 8.22T + 47T^{2} \)
53 \( 1 + 11.9iT - 53T^{2} \)
59 \( 1 - 11.8iT - 59T^{2} \)
61 \( 1 - 12.3iT - 61T^{2} \)
67 \( 1 - 15.2iT - 67T^{2} \)
71 \( 1 + 5.99T + 71T^{2} \)
73 \( 1 + 7.94T + 73T^{2} \)
79 \( 1 - 9.39T + 79T^{2} \)
83 \( 1 - 6.14iT - 83T^{2} \)
89 \( 1 - 13.0T + 89T^{2} \)
97 \( 1 + 16.0T + 97T^{2} \)
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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

−10.30497682366907795577249150202, −9.783073390739182388449923556155, −9.017374219568770636305907586136, −8.464670995477826905636614479322, −7.01601546707299374203052807423, −6.66247601369384901205505912067, −4.82845771779694297882760863963, −4.22683698041339427756126461329, −2.92074849493153440084112045892, −1.57677597976768952960575254293, 0.47939998797476906639493359595, 1.98725759977634000421845033495, 3.13283256014129503930133859343, 4.86543308696078842472956634940, 6.35402407009866414435846099645, 6.47399777790544888197401310256, 7.53361890034937686248603001834, 8.249945505070291209097928952037, 9.109538725858007621323510039207, 10.03409598590256988325451691304

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