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Properties

Label 2-760-8.5-c1-0-42
Degree $2$
Conductor $760$
Sign $0.101 - 0.994i$
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  + (−0.0481 + 1.41i)2-s + 1.80i·3-s + (−1.99 − 0.136i)4-s − i·5-s + (−2.55 − 0.0868i)6-s + 4.63·7-s + (0.288 − 2.81i)8-s − 0.260·9-s + (1.41 + 0.0481i)10-s − 5.12i·11-s + (0.245 − 3.60i)12-s + 0.771i·13-s + (−0.223 + 6.55i)14-s + 1.80·15-s + (3.96 + 0.542i)16-s + 7.88·17-s + ⋯
L(s)  = 1  + (−0.0340 + 0.999i)2-s + 1.04i·3-s + (−0.997 − 0.0680i)4-s − 0.447i·5-s + (−1.04 − 0.0354i)6-s + 1.75·7-s + (0.101 − 0.994i)8-s − 0.0868·9-s + (0.446 + 0.0152i)10-s − 1.54i·11-s + (0.0708 − 1.04i)12-s + 0.214i·13-s + (−0.0596 + 1.75i)14-s + 0.466·15-s + (0.990 + 0.135i)16-s + 1.91·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.101 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.101 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $0.101 - 0.994i$
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: \((2,\ 760,\ (\ :1/2),\ 0.101 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29279 + 1.16712i\)
\(L(\frac12)\) \(\approx\) \(1.29279 + 1.16712i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0481 - 1.41i)T \)
5 \( 1 + iT \)
19 \( 1 + iT \)
good3 \( 1 - 1.80iT - 3T^{2} \)
7 \( 1 - 4.63T + 7T^{2} \)
11 \( 1 + 5.12iT - 11T^{2} \)
13 \( 1 - 0.771iT - 13T^{2} \)
17 \( 1 - 7.88T + 17T^{2} \)
23 \( 1 - 1.39T + 23T^{2} \)
29 \( 1 + 6.35iT - 29T^{2} \)
31 \( 1 + 6.78T + 31T^{2} \)
37 \( 1 + 5.45iT - 37T^{2} \)
41 \( 1 + 6.96T + 41T^{2} \)
43 \( 1 + 1.29iT - 43T^{2} \)
47 \( 1 + 8.44T + 47T^{2} \)
53 \( 1 + 1.75iT - 53T^{2} \)
59 \( 1 - 5.88iT - 59T^{2} \)
61 \( 1 - 14.0iT - 61T^{2} \)
67 \( 1 - 9.05iT - 67T^{2} \)
71 \( 1 - 4.63T + 71T^{2} \)
73 \( 1 + 3.15T + 73T^{2} \)
79 \( 1 - 1.28T + 79T^{2} \)
83 \( 1 - 9.44iT - 83T^{2} \)
89 \( 1 - 3.77T + 89T^{2} \)
97 \( 1 + 5.71T + 97T^{2} \)
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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

−10.39351388880636436957087020649, −9.507871278559247361736301437482, −8.640575914791010336233316010144, −8.118165896157281909561755568608, −7.28045196511660716907085977640, −5.63981651132920732023349553863, −5.35538347909763770511507077343, −4.36869145606800939553708698067, −3.54403616923735711846080737778, −1.17861772057116507409544460501, 1.46031354562552306355551074854, 1.86992004043510032488339265924, 3.35183088877349145073679294897, 4.70575958033117414719083208776, 5.35116949093056796078442921256, 6.93412870538218344025376803099, 7.79742846027218046757046251003, 8.129698218715769112496923174052, 9.516439150346862675225986944312, 10.28066668809285113589518069975

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