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Properties

Label 2-756-63.59-c1-0-1
Degree $2$
Conductor $756$
Sign $-0.487 - 0.873i$
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  − 0.699·5-s + (−0.461 + 2.60i)7-s + 0.265i·11-s + (−1.13 − 0.657i)13-s + (−1.86 + 3.22i)17-s + (−0.382 + 0.220i)19-s + 4.96i·23-s − 4.51·25-s + (0.273 − 0.157i)29-s + (−4.85 + 2.80i)31-s + (0.322 − 1.82i)35-s + (−0.351 − 0.608i)37-s + (−5.39 + 9.34i)41-s + (3.73 + 6.46i)43-s + (3.50 − 6.06i)47-s + ⋯
L(s)  = 1  − 0.312·5-s + (−0.174 + 0.984i)7-s + 0.0799i·11-s + (−0.315 − 0.182i)13-s + (−0.452 + 0.783i)17-s + (−0.0877 + 0.0506i)19-s + 1.03i·23-s − 0.902·25-s + (0.0507 − 0.0292i)29-s + (−0.872 + 0.503i)31-s + (0.0545 − 0.308i)35-s + (−0.0577 − 0.0999i)37-s + (−0.842 + 1.45i)41-s + (0.569 + 0.985i)43-s + (0.510 − 0.884i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.487 - 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.487 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.487 - 0.873i$
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: \((2,\ 756,\ (\ :1/2),\ -0.487 - 0.873i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.443788 + 0.756086i\)
\(L(\frac12)\) \(\approx\) \(0.443788 + 0.756086i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.461 - 2.60i)T \)
good5 \( 1 + 0.699T + 5T^{2} \)
11 \( 1 - 0.265iT - 11T^{2} \)
13 \( 1 + (1.13 + 0.657i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.86 - 3.22i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.382 - 0.220i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.96iT - 23T^{2} \)
29 \( 1 + (-0.273 + 0.157i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.85 - 2.80i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.351 + 0.608i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.39 - 9.34i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.73 - 6.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.50 + 6.06i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-8.51 - 4.91i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.73 - 11.6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.89 + 2.82i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.97 - 5.14i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.4iT - 71T^{2} \)
73 \( 1 + (6.66 + 3.84i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.698 - 1.20i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.72 + 6.45i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.59 - 9.68i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.18 + 5.30i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.61815206648131484068813245495, −9.699607855726992924669447490516, −8.922166927107013255859915964335, −8.113701742793539600921084364488, −7.21832408943420340908835512018, −6.11503905407851908390555937656, −5.38238654849316955757298767943, −4.19481792827541059342720077185, −3.07940968862495292102468230196, −1.83483047714311549330654398187, 0.42997152524882187893442896042, 2.25259846550042582501883753336, 3.65902574591780933933021284065, 4.43332705425778612427613941561, 5.55597548532853086312247591872, 6.80774497337970798446368101082, 7.31625732983857660096986122422, 8.317821282130908727242259640211, 9.255101283294269588563323573430, 10.12320953860040027332848287928

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