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Properties

Label 2-756-63.59-c1-0-3
Degree $2$
Conductor $756$
Sign $0.990 + 0.135i$
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  + 1.68·5-s + (−0.0236 − 2.64i)7-s + 3.90i·11-s + (5.24 + 3.02i)13-s + (−0.201 + 0.348i)17-s + (−0.145 + 0.0840i)19-s − 8.88i·23-s − 2.15·25-s + (6.15 − 3.55i)29-s + (5.44 − 3.14i)31-s + (−0.0398 − 4.45i)35-s + (3.13 + 5.42i)37-s + (1.64 − 2.85i)41-s + (1.80 + 3.12i)43-s + (−4.38 + 7.59i)47-s + ⋯
L(s)  = 1  + 0.753·5-s + (−0.00893 − 0.999i)7-s + 1.17i·11-s + (1.45 + 0.839i)13-s + (−0.0488 + 0.0845i)17-s + (−0.0334 + 0.0192i)19-s − 1.85i·23-s − 0.431·25-s + (1.14 − 0.659i)29-s + (0.977 − 0.564i)31-s + (−0.00673 − 0.753i)35-s + (0.514 + 0.891i)37-s + (0.257 − 0.445i)41-s + (0.275 + 0.476i)43-s + (−0.639 + 1.10i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.135i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.990 + 0.135i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.990 + 0.135i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.84273 - 0.125208i\)
\(L(\frac12)\) \(\approx\) \(1.84273 - 0.125208i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0236 + 2.64i)T \)
good5 \( 1 - 1.68T + 5T^{2} \)
11 \( 1 - 3.90iT - 11T^{2} \)
13 \( 1 + (-5.24 - 3.02i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.201 - 0.348i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.145 - 0.0840i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 8.88iT - 23T^{2} \)
29 \( 1 + (-6.15 + 3.55i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.44 + 3.14i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.13 - 5.42i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.64 + 2.85i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.80 - 3.12i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.38 - 7.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.94 - 2.85i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.25 + 3.89i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.43 - 2.56i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.95 - 5.11i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.4iT - 71T^{2} \)
73 \( 1 + (6.05 + 3.49i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.603 - 1.04i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.181 - 0.314i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.38 + 2.39i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.508 + 0.293i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.21562595794372667677041365292, −9.679158621767202717709375622726, −8.629819694793521840612856762066, −7.76115351228554741159778298412, −6.54659399212115705711028104090, −6.26878559329586200148849004807, −4.64108853747998861659816735014, −4.10558892510188599470829793719, −2.50577125924326467180562041083, −1.24834101082009947887018131078, 1.29694501533217975501796370506, 2.75784524329797972433166972008, 3.67020764140072264437845882773, 5.40748904584465433519749803521, 5.74788569698840557379459514462, 6.61883792621157219040823601447, 8.082939182127536257424333978678, 8.627821188885510894511560351739, 9.441637508043274777810071639751, 10.35525680605348223838854943689

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