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Properties

Label 2-760-8.5-c1-0-64
Degree $2$
Conductor $760$
Sign $-0.801 - 0.597i$
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  + (−1.38 − 0.299i)2-s − 2.26i·3-s + (1.82 + 0.828i)4-s − i·5-s + (−0.678 + 3.12i)6-s − 1.28·7-s + (−2.26 − 1.68i)8-s − 2.12·9-s + (−0.299 + 1.38i)10-s − 6.52i·11-s + (1.87 − 4.12i)12-s + 5.04i·13-s + (1.77 + 0.384i)14-s − 2.26·15-s + (2.62 + 3.01i)16-s − 3.98·17-s + ⋯
L(s)  = 1  + (−0.977 − 0.211i)2-s − 1.30i·3-s + (0.910 + 0.414i)4-s − 0.447i·5-s + (−0.276 + 1.27i)6-s − 0.484·7-s + (−0.801 − 0.597i)8-s − 0.708·9-s + (−0.0947 + 0.437i)10-s − 1.96i·11-s + (0.541 − 1.18i)12-s + 1.39i·13-s + (0.473 + 0.102i)14-s − 0.584·15-s + (0.657 + 0.753i)16-s − 0.966·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 - 0.597i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.801 - 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $-0.801 - 0.597i$
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: \((2,\ 760,\ (\ :1/2),\ -0.801 - 0.597i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.149926 + 0.452132i\)
\(L(\frac12)\) \(\approx\) \(0.149926 + 0.452132i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 + 0.299i)T \)
5 \( 1 + iT \)
19 \( 1 + iT \)
good3 \( 1 + 2.26iT - 3T^{2} \)
7 \( 1 + 1.28T + 7T^{2} \)
11 \( 1 + 6.52iT - 11T^{2} \)
13 \( 1 - 5.04iT - 13T^{2} \)
17 \( 1 + 3.98T + 17T^{2} \)
23 \( 1 + 3.63T + 23T^{2} \)
29 \( 1 + 0.168iT - 29T^{2} \)
31 \( 1 + 3.93T + 31T^{2} \)
37 \( 1 + 9.41iT - 37T^{2} \)
41 \( 1 - 8.23T + 41T^{2} \)
43 \( 1 - 9.44iT - 43T^{2} \)
47 \( 1 + 3.01T + 47T^{2} \)
53 \( 1 - 7.82iT - 53T^{2} \)
59 \( 1 - 2.25iT - 59T^{2} \)
61 \( 1 - 3.69iT - 61T^{2} \)
67 \( 1 + 12.0iT - 67T^{2} \)
71 \( 1 - 1.36T + 71T^{2} \)
73 \( 1 + 9.86T + 73T^{2} \)
79 \( 1 - 14.7T + 79T^{2} \)
83 \( 1 - 6.95iT - 83T^{2} \)
89 \( 1 + 17.6T + 89T^{2} \)
97 \( 1 - 3.94T + 97T^{2} \)
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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

−9.474791079431210499903141485123, −8.934486644016583883144074416426, −8.167674030435701436378005572489, −7.34560558098709001992711686799, −6.40586130150107192234867251204, −5.98442489177108912470031109760, −4.01049307581417467213283845739, −2.67163868848234100129934614775, −1.56944940011450875060780900056, −0.32018680305922310483599005057, 2.12909397495295400116872258101, 3.34350799697073299949693710289, 4.54323052421887874453393945212, 5.55005176431401332085039913018, 6.68381980711592026925561197477, 7.45347365498116490794460716687, 8.439289237601933789366584102975, 9.507990326683261020264286120893, 9.954601224538915456145525363136, 10.40828993078677405201602703773

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