Properties

Label 2-760-1.1-c1-0-4
Degree $2$
Conductor $760$
Sign $1$
Analytic cond. $6.06863$
Root an. cond. $2.46345$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.642·3-s + 5-s + 3.58·7-s − 2.58·9-s + 1.35·13-s − 0.642·15-s + 5.58·17-s + 19-s − 2.30·21-s − 4.87·23-s + 25-s + 3.58·27-s + 9.58·29-s − 7.17·31-s + 3.58·35-s − 0.945·37-s − 0.871·39-s + 10.4·41-s − 2.71·43-s − 2.58·45-s − 5.89·47-s + 5.87·49-s − 3.58·51-s + 9.81·53-s − 0.642·57-s + 10.1·59-s + 3.28·61-s + ⋯
L(s)  = 1  − 0.370·3-s + 0.447·5-s + 1.35·7-s − 0.862·9-s + 0.376·13-s − 0.165·15-s + 1.35·17-s + 0.229·19-s − 0.502·21-s − 1.01·23-s + 0.200·25-s + 0.690·27-s + 1.78·29-s − 1.28·31-s + 0.606·35-s − 0.155·37-s − 0.139·39-s + 1.63·41-s − 0.414·43-s − 0.385·45-s − 0.859·47-s + 0.838·49-s − 0.502·51-s + 1.34·53-s − 0.0850·57-s + 1.32·59-s + 0.420·61-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(1.658660597\)
\(L(\frac12)\) \(\approx\) \(1.658660597\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
19 \( 1 - T \)
good3 \( 1 + 0.642T + 3T^{2} \)
7 \( 1 - 3.58T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 1.35T + 13T^{2} \)
17 \( 1 - 5.58T + 17T^{2} \)
23 \( 1 + 4.87T + 23T^{2} \)
29 \( 1 - 9.58T + 29T^{2} \)
31 \( 1 + 7.17T + 31T^{2} \)
37 \( 1 + 0.945T + 37T^{2} \)
41 \( 1 - 10.4T + 41T^{2} \)
43 \( 1 + 2.71T + 43T^{2} \)
47 \( 1 + 5.89T + 47T^{2} \)
53 \( 1 - 9.81T + 53T^{2} \)
59 \( 1 - 10.1T + 59T^{2} \)
61 \( 1 - 3.28T + 61T^{2} \)
67 \( 1 - 10.3T + 67T^{2} \)
71 \( 1 - 14.3T + 71T^{2} \)
73 \( 1 + 4.15T + 73T^{2} \)
79 \( 1 - 1.28T + 79T^{2} \)
83 \( 1 + 11.1T + 83T^{2} \)
89 \( 1 + 6.45T + 89T^{2} \)
97 \( 1 + 13.4T + 97T^{2} \)
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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

−10.44968139970453081226986153690, −9.560458463666220094475500346747, −8.389629143354915464958881966870, −8.021247988585065163288621818349, −6.77035964981026072585441356101, −5.62868764048575192137593915729, −5.24576155296133299179287079208, −3.95142913392650859996777779527, −2.55099828486234539448364061704, −1.20344395864318515258100406463, 1.20344395864318515258100406463, 2.55099828486234539448364061704, 3.95142913392650859996777779527, 5.24576155296133299179287079208, 5.62868764048575192137593915729, 6.77035964981026072585441356101, 8.021247988585065163288621818349, 8.389629143354915464958881966870, 9.560458463666220094475500346747, 10.44968139970453081226986153690

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