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Properties

Label 2-756-12.11-c1-0-39
Degree $2$
Conductor $756$
Sign $0.653 + 0.757i$
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  + (0.492 + 1.32i)2-s + (−1.51 + 1.30i)4-s − 3.62i·5-s + i·7-s + (−2.47 − 1.36i)8-s + (4.81 − 1.78i)10-s + 0.830·11-s − 4.86·13-s + (−1.32 + 0.492i)14-s + (0.586 − 3.95i)16-s − 3.28i·17-s − 6.45i·19-s + (4.74 + 5.49i)20-s + (0.409 + 1.10i)22-s + 5.08·23-s + ⋯
L(s)  = 1  + (0.348 + 0.937i)2-s + (−0.757 + 0.653i)4-s − 1.62i·5-s + 0.377i·7-s + (−0.876 − 0.482i)8-s + (1.52 − 0.565i)10-s + 0.250·11-s − 1.35·13-s + (−0.354 + 0.131i)14-s + (0.146 − 0.989i)16-s − 0.796i·17-s − 1.48i·19-s + (1.06 + 1.22i)20-s + (0.0872 + 0.234i)22-s + 1.06·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.653 + 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.653 + 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.653 + 0.757i$
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: \((2,\ 756,\ (\ :1/2),\ 0.653 + 0.757i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07843 - 0.493915i\)
\(L(\frac12)\) \(\approx\) \(1.07843 - 0.493915i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.492 - 1.32i)T \)
3 \( 1 \)
7 \( 1 - iT \)
good5 \( 1 + 3.62iT - 5T^{2} \)
11 \( 1 - 0.830T + 11T^{2} \)
13 \( 1 + 4.86T + 13T^{2} \)
17 \( 1 + 3.28iT - 17T^{2} \)
19 \( 1 + 6.45iT - 19T^{2} \)
23 \( 1 - 5.08T + 23T^{2} \)
29 \( 1 + 2.02iT - 29T^{2} \)
31 \( 1 + 7.12iT - 31T^{2} \)
37 \( 1 + 4.68T + 37T^{2} \)
41 \( 1 + 2.95iT - 41T^{2} \)
43 \( 1 - 1.62iT - 43T^{2} \)
47 \( 1 - 6.41T + 47T^{2} \)
53 \( 1 + 4.22iT - 53T^{2} \)
59 \( 1 + 8.31T + 59T^{2} \)
61 \( 1 + 5.55T + 61T^{2} \)
67 \( 1 - 11.6iT - 67T^{2} \)
71 \( 1 - 7.15T + 71T^{2} \)
73 \( 1 - 13.9T + 73T^{2} \)
79 \( 1 - 10.0iT - 79T^{2} \)
83 \( 1 + 16.5T + 83T^{2} \)
89 \( 1 - 15.9iT - 89T^{2} \)
97 \( 1 - 8.59T + 97T^{2} \)
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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

−9.626627687661284711052596201599, −9.247898216983993183898103330295, −8.545058053980451678307766670975, −7.58085061886232916793175415877, −6.80260465654797590784424938597, −5.46952108077223300435838665706, −4.98409243869683692152095395623, −4.24929433659206602124203158152, −2.62963886797682287653926018501, −0.53677109502352993773625973409, 1.77861974234294629659830335244, 2.97813185317328336891096993745, 3.65943512422024965872021034010, 4.86308114033661205187311247378, 6.02691856495026880529277127698, 6.90273717466561625641748092899, 7.78500277906961323501986288729, 9.050309965144881210610819600054, 10.07458003170034908585719618828, 10.48708980136452351921336658605

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