Properties

Label 2-722-361.139-c1-0-8
Degree $2$
Conductor $722$
Sign $0.998 - 0.0568i$
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  + (−0.991 − 0.128i)2-s + (0.859 − 1.26i)3-s + (0.967 + 0.254i)4-s + (0.0119 + 1.29i)5-s + (−1.01 + 1.14i)6-s + (−1.19 − 0.562i)7-s + (−0.926 − 0.376i)8-s + (0.244 + 0.617i)9-s + (0.154 − 1.28i)10-s + (0.00993 + 0.00610i)11-s + (1.15 − 1.00i)12-s + (0.289 + 2.85i)13-s + (1.11 + 0.711i)14-s + (1.64 + 1.09i)15-s + (0.870 + 0.492i)16-s + (2.50 + 2.52i)17-s + ⋯
L(s)  = 1  + (−0.701 − 0.0906i)2-s + (0.496 − 0.729i)3-s + (0.483 + 0.127i)4-s + (0.00532 + 0.580i)5-s + (−0.414 + 0.466i)6-s + (−0.450 − 0.212i)7-s + (−0.327 − 0.133i)8-s + (0.0813 + 0.205i)9-s + (0.0488 − 0.407i)10-s + (0.00299 + 0.00184i)11-s + (0.332 − 0.289i)12-s + (0.0801 + 0.790i)13-s + (0.296 + 0.190i)14-s + (0.426 + 0.283i)15-s + (0.217 + 0.123i)16-s + (0.607 + 0.612i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0568i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $0.998 - 0.0568i$
Analytic conductor: \(5.76519\)
Root analytic conductor: \(2.40108\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{722} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 722,\ (\ :1/2),\ 0.998 - 0.0568i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29213 + 0.0367690i\)
\(L(\frac12)\) \(\approx\) \(1.29213 + 0.0367690i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.991 + 0.128i)T \)
19 \( 1 + (-2.05 + 3.84i)T \)
good3 \( 1 + (-0.859 + 1.26i)T + (-1.10 - 2.78i)T^{2} \)
5 \( 1 + (-0.0119 - 1.29i)T + (-4.99 + 0.0918i)T^{2} \)
7 \( 1 + (1.19 + 0.562i)T + (4.45 + 5.40i)T^{2} \)
11 \( 1 + (-0.00993 - 0.00610i)T + (4.96 + 9.81i)T^{2} \)
13 \( 1 + (-0.289 - 2.85i)T + (-12.7 + 2.60i)T^{2} \)
17 \( 1 + (-2.50 - 2.52i)T + (-0.156 + 16.9i)T^{2} \)
23 \( 1 + (-5.06 + 0.372i)T + (22.7 - 3.36i)T^{2} \)
29 \( 1 + (2.63 - 3.72i)T + (-9.66 - 27.3i)T^{2} \)
31 \( 1 + (-0.797 - 2.54i)T + (-25.4 + 17.6i)T^{2} \)
37 \( 1 + (0.539 - 0.291i)T + (20.2 - 30.9i)T^{2} \)
41 \( 1 + (0.288 - 0.301i)T + (-1.88 - 40.9i)T^{2} \)
43 \( 1 + (-0.403 - 0.172i)T + (29.6 + 31.0i)T^{2} \)
47 \( 1 + (-2.13 - 0.437i)T + (43.2 + 18.4i)T^{2} \)
53 \( 1 + (-2.28 - 6.86i)T + (-42.4 + 31.7i)T^{2} \)
59 \( 1 + (0.761 + 2.60i)T + (-49.6 + 31.8i)T^{2} \)
61 \( 1 + (-7.34 + 6.63i)T + (6.15 - 60.6i)T^{2} \)
67 \( 1 + (10.3 + 0.379i)T + (66.8 + 4.91i)T^{2} \)
71 \( 1 + (-6.26 - 5.66i)T + (7.16 + 70.6i)T^{2} \)
73 \( 1 + (-0.745 + 2.73i)T + (-62.8 - 37.0i)T^{2} \)
79 \( 1 + (-9.01 + 6.75i)T + (22.1 - 75.8i)T^{2} \)
83 \( 1 + (-3.32 + 8.87i)T + (-62.5 - 54.5i)T^{2} \)
89 \( 1 + (-3.93 - 14.4i)T + (-76.6 + 45.2i)T^{2} \)
97 \( 1 + (0.731 - 0.0268i)T + (96.7 - 7.12i)T^{2} \)
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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

−10.48980701439373158450718680666, −9.407215201258317601560410151405, −8.736634627508386755314478359738, −7.74935293873035359291003924807, −7.00228914954937474328962064296, −6.53821456044845410591917797464, −5.03660247569943817538964479732, −3.46222925808765207309964125017, −2.54571512659550952810328807057, −1.29038327217683856701420725473, 0.955590930056378053564648925373, 2.79924379186590484934213183691, 3.69113711508825466728629497642, 5.00885730146910988238916841110, 5.92544417485124977853737795241, 7.10285045445759553997985165676, 8.055189522159592422760545825156, 8.842024936866256627934877205640, 9.559612299628738689804500285932, 10.02166883656768431737474912048

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