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Properties

Label 2-760-8.5-c1-0-71
Degree $2$
Conductor $760$
Sign $-0.642 - 0.766i$
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  + (1.02 − 0.970i)2-s − 2.59i·3-s + (0.116 − 1.99i)4-s + i·5-s + (−2.52 − 2.67i)6-s − 4.55·7-s + (−1.81 − 2.16i)8-s − 3.74·9-s + (0.970 + 1.02i)10-s + 4.44i·11-s + (−5.18 − 0.302i)12-s − 4.58i·13-s + (−4.68 + 4.41i)14-s + 2.59·15-s + (−3.97 − 0.465i)16-s + 0.771·17-s + ⋯
L(s)  = 1  + (0.727 − 0.686i)2-s − 1.49i·3-s + (0.0583 − 0.998i)4-s + 0.447i·5-s + (−1.02 − 1.09i)6-s − 1.72·7-s + (−0.642 − 0.766i)8-s − 1.24·9-s + (0.306 + 0.325i)10-s + 1.34i·11-s + (−1.49 − 0.0874i)12-s − 1.27i·13-s + (−1.25 + 1.18i)14-s + 0.670·15-s + (−0.993 − 0.116i)16-s + 0.187·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $-0.642 - 0.766i$
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: \((2,\ 760,\ (\ :1/2),\ -0.642 - 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.484417 + 1.03848i\)
\(L(\frac12)\) \(\approx\) \(0.484417 + 1.03848i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.02 + 0.970i)T \)
5 \( 1 - iT \)
19 \( 1 - iT \)
good3 \( 1 + 2.59iT - 3T^{2} \)
7 \( 1 + 4.55T + 7T^{2} \)
11 \( 1 - 4.44iT - 11T^{2} \)
13 \( 1 + 4.58iT - 13T^{2} \)
17 \( 1 - 0.771T + 17T^{2} \)
23 \( 1 - 1.32T + 23T^{2} \)
29 \( 1 + 9.02iT - 29T^{2} \)
31 \( 1 + 8.26T + 31T^{2} \)
37 \( 1 + 3.29iT - 37T^{2} \)
41 \( 1 - 7.00T + 41T^{2} \)
43 \( 1 + 11.3iT - 43T^{2} \)
47 \( 1 + 1.75T + 47T^{2} \)
53 \( 1 - 6.73iT - 53T^{2} \)
59 \( 1 - 10.5iT - 59T^{2} \)
61 \( 1 + 11.0iT - 61T^{2} \)
67 \( 1 + 10.0iT - 67T^{2} \)
71 \( 1 - 4.39T + 71T^{2} \)
73 \( 1 + 4.07T + 73T^{2} \)
79 \( 1 + 11.4T + 79T^{2} \)
83 \( 1 - 10.9iT - 83T^{2} \)
89 \( 1 - 2.17T + 89T^{2} \)
97 \( 1 - 5.66T + 97T^{2} \)
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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

−9.974945329967048126535135132862, −9.218741133807744321221063651014, −7.62691287669572361158061031843, −7.06682064308496196274277575023, −6.21732356625236330840934432183, −5.59210267249247857134612603734, −3.92568127369055676573951422280, −2.89066622468808868741454116225, −2.07459124505121041282171662970, −0.43044667421565693895826996872, 3.08126155070653554936470840471, 3.59976515142978990531817505469, 4.52087364665550023238749498073, 5.50461203893265627139200722756, 6.25557139331879055555732641320, 7.15440648844709510859121675119, 8.687859521874912428249704587292, 9.082909582156243805042537544695, 9.793219767615003742505826202865, 10.93870205292114358071867005663

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