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Properties

Label 2-756-7.4-c1-0-3
Degree $2$
Conductor $756$
Sign $0.386 - 0.922i$
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  + (1.58 + 2.73i)5-s + (0.5 + 2.59i)7-s + (3.16 − 5.47i)11-s + (−1.58 + 2.73i)17-s + (3.5 + 6.06i)19-s + (−1.58 − 2.73i)23-s + (−2.5 + 4.33i)25-s − 3.16·29-s + (−1.5 + 2.59i)31-s + (−6.32 + 5.47i)35-s + (2 + 3.46i)37-s − 9.48·41-s + 5·43-s + (4.74 + 8.21i)47-s + (−6.5 + 2.59i)49-s + ⋯
L(s)  = 1  + (0.707 + 1.22i)5-s + (0.188 + 0.981i)7-s + (0.953 − 1.65i)11-s + (−0.383 + 0.664i)17-s + (0.802 + 1.39i)19-s + (−0.329 − 0.571i)23-s + (−0.5 + 0.866i)25-s − 0.587·29-s + (−0.269 + 0.466i)31-s + (−1.06 + 0.925i)35-s + (0.328 + 0.569i)37-s − 1.48·41-s + 0.762·43-s + (0.691 + 1.19i)47-s + (−0.928 + 0.371i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49485 + 0.994348i\)
\(L(\frac12)\) \(\approx\) \(1.49485 + 0.994348i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 2.59i)T \)
good5 \( 1 + (-1.58 - 2.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.16 + 5.47i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (1.58 - 2.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.5 - 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.58 + 2.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.16T + 29T^{2} \)
31 \( 1 + (1.5 - 2.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 9.48T + 41T^{2} \)
43 \( 1 - 5T + 43T^{2} \)
47 \( 1 + (-4.74 - 8.21i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.74 + 8.21i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.32 + 10.9i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.5 - 2.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5 - 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.6T + 71T^{2} \)
73 \( 1 + (-2.5 + 4.33i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6 + 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.32T + 83T^{2} \)
89 \( 1 + (4.74 + 8.21i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5T + 97T^{2} \)
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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

−10.52623922907704644511410382389, −9.709472099379997442981094330745, −8.775272843372157141915392666825, −8.088285676675425195882868649365, −6.74180495241961951572975876537, −6.07446430343482198848618268123, −5.52817429289512831921921649265, −3.77315166425405347126678405918, −2.93625720098955296498880892687, −1.71683730666216784679185480171, 1.01078842271763297194305842871, 2.12496372588200648486040085856, 3.98728877621667091530523756105, 4.71810395129548834197003327002, 5.50777035672276479235371154492, 6.96243514482671641660078505422, 7.34219447592412128936471237106, 8.699186833030610670134219600438, 9.511239390591076959564535976013, 9.797550949698101585655651859205

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