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Properties

Label 2-756-12.11-c1-0-5
Degree $2$
Conductor $756$
Sign $0.181 - 0.983i$
Analytic cond. $6.03669$
Root an. cond. $2.45696$
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  + (−0.904 − 1.08i)2-s + (−0.363 + 1.96i)4-s + 2.20i·5-s + i·7-s + (2.46 − 1.38i)8-s + (2.39 − 1.99i)10-s − 2.61·11-s + 4.38·13-s + (1.08 − 0.904i)14-s + (−3.73 − 1.42i)16-s − 5.03i·17-s + 6.26i·19-s + (−4.33 − 0.801i)20-s + (2.36 + 2.84i)22-s − 5.55·23-s + ⋯
L(s)  = 1  + (−0.639 − 0.768i)2-s + (−0.181 + 0.983i)4-s + 0.986i·5-s + 0.377i·7-s + (0.871 − 0.489i)8-s + (0.758 − 0.631i)10-s − 0.789·11-s + 1.21·13-s + (0.290 − 0.241i)14-s + (−0.934 − 0.357i)16-s − 1.22i·17-s + 1.43i·19-s + (−0.970 − 0.179i)20-s + (0.505 + 0.606i)22-s − 1.15·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.181 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.181 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.181 - 0.983i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.181 - 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.605373 + 0.503851i\)
\(L(\frac12)\) \(\approx\) \(0.605373 + 0.503851i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.904 + 1.08i)T \)
3 \( 1 \)
7 \( 1 - iT \)
good5 \( 1 - 2.20iT - 5T^{2} \)
11 \( 1 + 2.61T + 11T^{2} \)
13 \( 1 - 4.38T + 13T^{2} \)
17 \( 1 + 5.03iT - 17T^{2} \)
19 \( 1 - 6.26iT - 19T^{2} \)
23 \( 1 + 5.55T + 23T^{2} \)
29 \( 1 - 7.73iT - 29T^{2} \)
31 \( 1 - 7.01iT - 31T^{2} \)
37 \( 1 + 8.26T + 37T^{2} \)
41 \( 1 - 0.297iT - 41T^{2} \)
43 \( 1 - 1.46iT - 43T^{2} \)
47 \( 1 + 5.24T + 47T^{2} \)
53 \( 1 - 11.3iT - 53T^{2} \)
59 \( 1 + 5.17T + 59T^{2} \)
61 \( 1 + 6.40T + 61T^{2} \)
67 \( 1 + 5.83iT - 67T^{2} \)
71 \( 1 - 7.44T + 71T^{2} \)
73 \( 1 - 10.5T + 73T^{2} \)
79 \( 1 - 1.43iT - 79T^{2} \)
83 \( 1 - 6.64T + 83T^{2} \)
89 \( 1 - 6.50iT - 89T^{2} \)
97 \( 1 + 16.5T + 97T^{2} \)
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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

−10.64659281273264872765279881441, −9.876487757632668956869264878239, −8.873239128748554402253592933732, −8.113039929534554515015516047091, −7.25600628629804087800616390340, −6.30746017772368628714212788617, −5.06935878372014450953240613914, −3.58456899902991638731608745121, −2.94641932355291715699213525987, −1.65146521739498898579971660544, 0.50274213781855906632335651927, 1.92240938381444609725522230025, 3.94601018048882922225592018844, 4.90169082420317691620255881140, 5.84622171850097183433846849629, 6.62458261435383991615177146210, 7.88578970866502230741600945629, 8.315561834408299534798885573424, 9.107740469761330198118036730342, 10.01770086904228054701472775182

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