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Properties

Label 2-790-395.372-c1-0-18
Degree $2$
Conductor $790$
Sign $0.972 + 0.233i$
Analytic cond. $6.30818$
Root an. cond. $2.51160$
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  + (0.258 − 0.965i)2-s + (0.160 − 0.597i)3-s + (−0.866 − 0.499i)4-s + (2.04 + 0.900i)5-s + (−0.535 − 0.309i)6-s + (0.491 + 1.83i)7-s + (−0.707 + 0.707i)8-s + (2.26 + 1.30i)9-s + (1.39 − 1.74i)10-s + (−1.45 + 2.51i)11-s + (−0.437 + 0.437i)12-s + (−1.44 − 0.386i)13-s + 1.89·14-s + (0.865 − 1.07i)15-s + (0.500 + 0.866i)16-s + (0.791 − 0.791i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (0.0924 − 0.344i)3-s + (−0.433 − 0.249i)4-s + (0.915 + 0.402i)5-s + (−0.218 − 0.126i)6-s + (0.185 + 0.692i)7-s + (−0.249 + 0.249i)8-s + (0.755 + 0.436i)9-s + (0.442 − 0.551i)10-s + (−0.437 + 0.757i)11-s + (−0.126 + 0.126i)12-s + (−0.400 − 0.107i)13-s + 0.507·14-s + (0.223 − 0.278i)15-s + (0.125 + 0.216i)16-s + (0.192 − 0.192i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.233i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 + 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(790\)    =    \(2 \cdot 5 \cdot 79\)
Sign: $0.972 + 0.233i$
Analytic conductor: \(6.30818\)
Root analytic conductor: \(2.51160\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{790} (767, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 790,\ (\ :1/2),\ 0.972 + 0.233i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.00119 - 0.237094i\)
\(L(\frac12)\) \(\approx\) \(2.00119 - 0.237094i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 + (-2.04 - 0.900i)T \)
79 \( 1 + (-5.07 - 7.29i)T \)
good3 \( 1 + (-0.160 + 0.597i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (-0.491 - 1.83i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (1.45 - 2.51i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.44 + 0.386i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-0.791 + 0.791i)T - 17iT^{2} \)
19 \( 1 + (-3.25 - 1.87i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.38 - 5.16i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (0.731 - 1.26i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.99 - 3.44i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.33 + 1.16i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 10.1iT - 41T^{2} \)
43 \( 1 + (-0.130 + 0.485i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (3.13 + 11.7i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-4.10 + 1.10i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-3.77 - 6.54i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 12.4iT - 61T^{2} \)
67 \( 1 + (3.82 + 3.82i)T + 67iT^{2} \)
71 \( 1 + 2.63iT - 71T^{2} \)
73 \( 1 + (2.60 + 9.73i)T + (-63.2 + 36.5i)T^{2} \)
83 \( 1 + (-3.03 - 11.3i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + 1.64iT - 89T^{2} \)
97 \( 1 + (-0.129 - 0.129i)T + 97iT^{2} \)
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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

−10.18963989717097578118326362411, −9.672259815668583823573381173916, −8.776952114119152507979988064032, −7.54473214571635038930106084305, −6.89788552353792194755799199015, −5.44463365415525414146253675479, −5.12566320744178612821378077245, −3.54816445653890155050180571888, −2.32993262678717682465368876562, −1.66005147207636395903401699475, 1.06309339582417537212152315139, 2.88054488334972449855854398621, 4.19623608867516705590716778654, 4.93991662863065193508233817137, 5.90876702821819265208673991723, 6.78586475288708323473942356036, 7.66841235689519554836523495671, 8.642525230444726845836942676129, 9.469531319404500337166370428259, 10.10489821252139551251637149121

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