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Properties

Label 2-722-361.140-c1-0-22
Degree $2$
Conductor $722$
Sign $0.312 + 0.949i$
Analytic cond. $5.76519$
Root an. cond. $2.40108$
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  + (0.926 + 0.376i)2-s + (−1.05 − 0.237i)3-s + (0.716 + 0.697i)4-s + (−0.0324 − 1.17i)5-s + (−0.891 − 0.618i)6-s + (0.291 + 1.15i)7-s + (0.401 + 0.915i)8-s + (−1.64 − 0.778i)9-s + (0.412 − 1.10i)10-s + (0.486 − 5.87i)11-s + (−0.593 − 0.908i)12-s + (−1.00 − 3.22i)13-s + (−0.163 + 1.17i)14-s + (−0.244 + 1.25i)15-s + (0.0275 + 0.999i)16-s + (−1.66 + 1.62i)17-s + ⋯
L(s)  = 1  + (0.655 + 0.266i)2-s + (−0.611 − 0.136i)3-s + (0.358 + 0.348i)4-s + (−0.0144 − 0.525i)5-s + (−0.363 − 0.252i)6-s + (0.110 + 0.434i)7-s + (0.142 + 0.323i)8-s + (−0.549 − 0.259i)9-s + (0.130 − 0.348i)10-s + (0.146 − 1.77i)11-s + (−0.171 − 0.262i)12-s + (−0.279 − 0.893i)13-s + (−0.0435 + 0.314i)14-s + (−0.0631 + 0.323i)15-s + (0.00688 + 0.249i)16-s + (−0.404 + 0.393i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.312 + 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.312 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $0.312 + 0.949i$
Analytic conductor: \(5.76519\)
Root analytic conductor: \(2.40108\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{722} (501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 722,\ (\ :1/2),\ 0.312 + 0.949i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20085 - 0.869129i\)
\(L(\frac12)\) \(\approx\) \(1.20085 - 0.869129i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.926 - 0.376i)T \)
19 \( 1 + (-3.23 + 2.92i)T \)
good3 \( 1 + (1.05 + 0.237i)T + (2.71 + 1.28i)T^{2} \)
5 \( 1 + (0.0324 + 1.17i)T + (-4.99 + 0.275i)T^{2} \)
7 \( 1 + (-0.291 - 1.15i)T + (-6.15 + 3.33i)T^{2} \)
11 \( 1 + (-0.486 + 5.87i)T + (-10.8 - 1.81i)T^{2} \)
13 \( 1 + (1.00 + 3.22i)T + (-10.6 + 7.40i)T^{2} \)
17 \( 1 + (1.66 - 1.62i)T + (0.468 - 16.9i)T^{2} \)
23 \( 1 + (2.51 - 0.563i)T + (20.8 - 9.81i)T^{2} \)
29 \( 1 + (-1.83 - 0.519i)T + (24.7 + 15.1i)T^{2} \)
31 \( 1 + (4.06 + 3.16i)T + (7.61 + 30.0i)T^{2} \)
37 \( 1 + (-0.868 + 10.4i)T + (-36.4 - 6.08i)T^{2} \)
41 \( 1 + (-3.72 - 3.24i)T + (5.63 + 40.6i)T^{2} \)
43 \( 1 + (1.38 + 3.70i)T + (-32.4 + 28.2i)T^{2} \)
47 \( 1 + (2.99 + 2.07i)T + (16.4 + 44.0i)T^{2} \)
53 \( 1 + (0.780 + 0.541i)T + (18.5 + 49.6i)T^{2} \)
59 \( 1 + (-2.16 - 1.88i)T + (8.10 + 58.4i)T^{2} \)
61 \( 1 + (1.03 + 1.41i)T + (-18.2 + 58.2i)T^{2} \)
67 \( 1 + (11.5 + 1.27i)T + (65.3 + 14.6i)T^{2} \)
71 \( 1 + (2.24 - 3.05i)T + (-21.1 - 67.7i)T^{2} \)
73 \( 1 + (4.84 - 4.71i)T + (2.01 - 72.9i)T^{2} \)
79 \( 1 + (-2.89 - 7.72i)T + (-59.5 + 51.8i)T^{2} \)
83 \( 1 + (4.00 - 2.16i)T + (45.3 - 69.4i)T^{2} \)
89 \( 1 + (-11.2 - 10.9i)T + (2.45 + 88.9i)T^{2} \)
97 \( 1 + (-11.8 + 1.30i)T + (94.6 - 21.2i)T^{2} \)
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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

−10.59087604119804601269233470258, −9.065156439960854251872910774480, −8.568574823075461714397698449902, −7.56004644019742951765116200727, −6.33261938826410387426250624600, −5.66785187630313871200776748569, −5.14381742272375348743690704916, −3.72014956274292000139738536157, −2.70925153671703578697985643575, −0.66820889975105075643900672843, 1.77662261583157809241880638115, 3.00008820712471508196131917520, 4.41850004979610374065775267507, 4.88460341041437057818693592446, 6.12225749212407151744916612513, 6.93976858494538360139546838919, 7.63103159129805893721308181668, 9.098723328746657995682575583207, 10.08752923851510103438857158469, 10.57733217087211793761293737918

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