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Properties

Label 2-722-19.11-c1-0-5
Degree $2$
Conductor $722$
Sign $0.715 - 0.699i$
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  + (0.5 − 0.866i)2-s + (0.642 − 1.11i)3-s + (−0.499 − 0.866i)4-s + (−1.84 + 3.20i)5-s + (−0.642 − 1.11i)6-s − 0.442·7-s − 0.999·8-s + (0.675 + 1.17i)9-s + (1.84 + 3.20i)10-s − 4.02·11-s − 1.28·12-s + (2.44 + 4.23i)13-s + (−0.221 + 0.383i)14-s + (2.37 + 4.10i)15-s + (−0.5 + 0.866i)16-s + (−0.133 + 0.231i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.370 − 0.642i)3-s + (−0.249 − 0.433i)4-s + (−0.826 + 1.43i)5-s + (−0.262 − 0.453i)6-s − 0.167·7-s − 0.353·8-s + (0.225 + 0.390i)9-s + (0.584 + 1.01i)10-s − 1.21·11-s − 0.370·12-s + (0.678 + 1.17i)13-s + (−0.0591 + 0.102i)14-s + (0.612 + 1.06i)15-s + (−0.125 + 0.216i)16-s + (−0.0323 + 0.0560i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.715 - 0.699i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.715 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $0.715 - 0.699i$
Analytic conductor: \(5.76519\)
Root analytic conductor: \(2.40108\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{722} (429, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 722,\ (\ :1/2),\ 0.715 - 0.699i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22409 + 0.498892i\)
\(L(\frac12)\) \(\approx\) \(1.22409 + 0.498892i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 \)
good3 \( 1 + (-0.642 + 1.11i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.84 - 3.20i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 0.442T + 7T^{2} \)
11 \( 1 + 4.02T + 11T^{2} \)
13 \( 1 + (-2.44 - 4.23i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.133 - 0.231i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-4.60 - 7.97i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.111 + 0.193i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.47T + 31T^{2} \)
37 \( 1 + 1.44T + 37T^{2} \)
41 \( 1 + (3.93 - 6.81i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.81 + 4.88i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.09 + 1.90i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.97 - 8.61i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.75 - 3.03i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.03 + 3.53i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.0738 - 0.127i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.73 + 9.93i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.711 - 1.23i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.10 + 8.83i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.28T + 83T^{2} \)
89 \( 1 + (2.98 + 5.16i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.04 + 3.53i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.82351393978392196391348989004, −9.910856050801155623607402154644, −8.754408038046278722482942410200, −7.64438993349688455379633087390, −7.20429823529934799411083293683, −6.23133571412429991441628178986, −4.90757373080489241902338045112, −3.65630169495947149407858674479, −2.90745297262201743047609901476, −1.81563354144600064224392784827, 0.58816523188898089264080613950, 3.03607563691157815910481447397, 3.98167861634377888319434181292, 4.85964658732148902695370296340, 5.49035455013960374315261034377, 6.83677093185945372574918448121, 8.003200025596852729197153917665, 8.442737086374511478658533109920, 9.151436872184619412394921637711, 10.20508189760664670377681757055

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