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Properties

Label 2-756-63.20-c1-0-7
Degree $2$
Conductor $756$
Sign $-0.826 - 0.563i$
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  + (−2.09 − 3.62i)5-s + (−2.64 − 0.0532i)7-s + (1.23 + 0.711i)11-s + (0.850 − 0.491i)13-s + 0.370·17-s + 4.97i·19-s + (−4.98 + 2.87i)23-s + (−6.26 + 10.8i)25-s + (−7.31 − 4.22i)29-s + (−6.28 + 3.62i)31-s + (5.34 + 9.70i)35-s + 3.46·37-s + (−1.06 − 1.85i)41-s + (3.00 − 5.21i)43-s + (−4.13 + 7.16i)47-s + ⋯
L(s)  = 1  + (−0.936 − 1.62i)5-s + (−0.999 − 0.0201i)7-s + (0.371 + 0.214i)11-s + (0.235 − 0.136i)13-s + 0.0899·17-s + 1.14i·19-s + (−1.04 + 0.600i)23-s + (−1.25 + 2.17i)25-s + (−1.35 − 0.784i)29-s + (−1.12 + 0.651i)31-s + (0.903 + 1.64i)35-s + 0.569·37-s + (−0.167 − 0.289i)41-s + (0.458 − 0.794i)43-s + (−0.603 + 1.04i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.826 - 0.563i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.826 - 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.826 - 0.563i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.826 - 0.563i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0225862 + 0.0732051i\)
\(L(\frac12)\) \(\approx\) \(0.0225862 + 0.0732051i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.64 + 0.0532i)T \)
good5 \( 1 + (2.09 + 3.62i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.23 - 0.711i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.850 + 0.491i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 0.370T + 17T^{2} \)
19 \( 1 - 4.97iT - 19T^{2} \)
23 \( 1 + (4.98 - 2.87i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (7.31 + 4.22i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (6.28 - 3.62i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.46T + 37T^{2} \)
41 \( 1 + (1.06 + 1.85i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.00 + 5.21i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.13 - 7.16i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 4.97iT - 53T^{2} \)
59 \( 1 + (2.27 + 3.94i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.50 + 3.75i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.03 - 8.71i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.9iT - 71T^{2} \)
73 \( 1 + 9.52iT - 73T^{2} \)
79 \( 1 + (-4.25 + 7.37i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.972 + 1.68i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 7.80T + 89T^{2} \)
97 \( 1 + (-3.34 - 1.92i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.476797553889763363204478978850, −9.177318116365395257236337080050, −8.040314888316435825876505687860, −7.52579995134338765107367562026, −6.13601012849669863417103521044, −5.34737121044815351862826403339, −4.09943052158667387267534020077, −3.59924712478486908902792559391, −1.58953717186067229139001731660, −0.03850738538987742577010391539, 2.48166239964308908120639597710, 3.43310773555680326797352617292, 4.10620391722852101255109103372, 5.81005120235595016742216225077, 6.69265229587168230312953089924, 7.17257603862867370195080256168, 8.145821593584283532649751619396, 9.276947641873699691479069564036, 10.06178055780047361574798379681, 11.04308022710147964631285332553

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