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Properties

Label 2-760-760.189-c0-0-1
Degree $2$
Conductor $760$
Sign $0.923 + 0.382i$
Analytic cond. $0.379289$
Root an. cond. $0.615864$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.382 − 0.923i)2-s − 0.765i·3-s + (−0.707 + 0.707i)4-s + i·5-s + (−0.707 + 0.292i)6-s + (0.923 + 0.382i)8-s + 0.414·9-s + (0.923 − 0.382i)10-s + 1.41i·11-s + (0.541 + 0.541i)12-s + 1.84i·13-s + 0.765·15-s − i·16-s + (−0.158 − 0.382i)18-s − i·19-s + (−0.707 − 0.707i)20-s + ⋯
L(s)  = 1  + (−0.382 − 0.923i)2-s − 0.765i·3-s + (−0.707 + 0.707i)4-s + i·5-s + (−0.707 + 0.292i)6-s + (0.923 + 0.382i)8-s + 0.414·9-s + (0.923 − 0.382i)10-s + 1.41i·11-s + (0.541 + 0.541i)12-s + 1.84i·13-s + 0.765·15-s − i·16-s + (−0.158 − 0.382i)18-s − i·19-s + (−0.707 − 0.707i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $0.923 + 0.382i$
Analytic conductor: \(0.379289\)
Root analytic conductor: \(0.615864\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{760} (189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 760,\ (\ :0),\ 0.923 + 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7652315986\)
\(L(\frac12)\) \(\approx\) \(0.7652315986\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.382 + 0.923i)T \)
5 \( 1 - iT \)
19 \( 1 + iT \)
good3 \( 1 + 0.765iT - T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - 1.41iT - T^{2} \)
13 \( 1 - 1.84iT - T^{2} \)
17 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + 0.765iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 0.765iT - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + 1.41iT - T^{2} \)
67 \( 1 + 1.84iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + 1.84T + T^{2} \)
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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

−10.55044517793837874624863700973, −9.645858487647965762154122222754, −9.118886733049526694471597905605, −7.73446317341803541229733829064, −7.12342606537913083977318859501, −6.55009617606950544689008705295, −4.71721524418332331860155676176, −3.92515368015775189451262716555, −2.42085619131412664265747652184, −1.78344252486711102387075959893, 1.04323228490519563664463447560, 3.44280843102116869893743109298, 4.43419599092851498938453175301, 5.51542614978338252233282019947, 5.81730222421585674546816566393, 7.32190560437348964866597178948, 8.327613216072979561010171038131, 8.592753052261509320041223596891, 9.774525656887856241351642908736, 10.22467616710998489542562032798

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