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Properties

Label 2-756-9.4-c1-0-3
Degree $2$
Conductor $756$
Sign $0.989 + 0.145i$
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  + (−0.555 + 0.962i)5-s + (−0.5 − 0.866i)7-s + (−0.944 − 1.63i)11-s + (0.5 − 0.866i)13-s + 5.87·17-s + 7.09·19-s + (1.99 − 3.45i)23-s + (1.88 + 3.26i)25-s + (0.493 + 0.855i)29-s + (0.333 − 0.576i)31-s + 1.11·35-s − 1.33·37-s + (−0.944 + 1.63i)41-s + (5.43 + 9.40i)43-s + (−5.54 − 9.61i)47-s + ⋯
L(s)  = 1  + (−0.248 + 0.430i)5-s + (−0.188 − 0.327i)7-s + (−0.284 − 0.493i)11-s + (0.138 − 0.240i)13-s + 1.42·17-s + 1.62·19-s + (0.415 − 0.720i)23-s + (0.376 + 0.652i)25-s + (0.0916 + 0.158i)29-s + (0.0598 − 0.103i)31-s + 0.187·35-s − 0.219·37-s + (−0.147 + 0.255i)41-s + (0.828 + 1.43i)43-s + (−0.809 − 1.40i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.145i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.989 + 0.145i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.989 + 0.145i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49614 - 0.109188i\)
\(L(\frac12)\) \(\approx\) \(1.49614 - 0.109188i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (0.555 - 0.962i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.944 + 1.63i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 5.87T + 17T^{2} \)
19 \( 1 - 7.09T + 19T^{2} \)
23 \( 1 + (-1.99 + 3.45i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.493 - 0.855i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.333 + 0.576i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.33T + 37T^{2} \)
41 \( 1 + (0.944 - 1.63i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.43 - 9.40i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.54 + 9.61i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 12.2T + 53T^{2} \)
59 \( 1 + (-2.38 + 4.12i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.88 + 3.26i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.04 - 3.54i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.7T + 71T^{2} \)
73 \( 1 + 3.09T + 73T^{2} \)
79 \( 1 + (3.21 + 5.56i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.93 + 10.2i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 14.3T + 89T^{2} \)
97 \( 1 + (0.382 + 0.662i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.28060994655989224768052021554, −9.643449559664113188462117705235, −8.529065751401570212675231171134, −7.65128364008286494720398604368, −6.99445962458058099268646734492, −5.83526474972632247702276043499, −5.02162545401017479031138444263, −3.57619716120214753988244862522, −2.94284488232426332824125288468, −1.01942786560611794143905654828, 1.16255006564115699392039085438, 2.78223821277526226933979136015, 3.86035194981524524969622361774, 5.10166165030897111845489729637, 5.71962756268206802314002087600, 7.05905396955844962605767671946, 7.73435118983721620045315696107, 8.685448195444007063152381128524, 9.569406022924669278628481486173, 10.17962826535728632116097585601

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