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Properties

Label 2-637-91.80-c1-0-35
Degree $2$
Conductor $637$
Sign $-0.999 - 0.0283i$
Analytic cond. $5.08647$
Root an. cond. $2.25532$
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  + (−0.433 − 1.61i)2-s − 0.637i·3-s + (−0.700 + 0.404i)4-s + (0.520 − 1.94i)5-s + (−1.03 + 0.276i)6-s + (−1.41 − 1.41i)8-s + 2.59·9-s − 3.36·10-s + (0.694 + 0.694i)11-s + (0.258 + 0.446i)12-s + (1.60 − 3.22i)13-s + (−1.23 − 0.331i)15-s + (−2.48 + 4.29i)16-s + (−2.99 − 5.18i)17-s + (−1.12 − 4.19i)18-s + (1.98 + 1.98i)19-s + ⋯
L(s)  = 1  + (−0.306 − 1.14i)2-s − 0.368i·3-s + (−0.350 + 0.202i)4-s + (0.232 − 0.868i)5-s + (−0.421 + 0.112i)6-s + (−0.498 − 0.498i)8-s + 0.864·9-s − 1.06·10-s + (0.209 + 0.209i)11-s + (0.0744 + 0.129i)12-s + (0.446 − 0.894i)13-s + (−0.319 − 0.0856i)15-s + (−0.620 + 1.07i)16-s + (−0.725 − 1.25i)17-s + (−0.265 − 0.989i)18-s + (0.455 + 0.455i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0283i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0283i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.999 - 0.0283i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.999 - 0.0283i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0192502 + 1.35631i\)
\(L(\frac12)\) \(\approx\) \(0.0192502 + 1.35631i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 + (-1.60 + 3.22i)T \)
good2 \( 1 + (0.433 + 1.61i)T + (-1.73 + i)T^{2} \)
3 \( 1 + 0.637iT - 3T^{2} \)
5 \( 1 + (-0.520 + 1.94i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.694 - 0.694i)T + 11iT^{2} \)
17 \( 1 + (2.99 + 5.18i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.98 - 1.98i)T + 19iT^{2} \)
23 \( 1 + (2.58 + 1.49i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.65 - 6.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (8.34 - 2.23i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-4.63 + 1.24i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.886 + 3.30i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (0.748 + 0.432i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.96 + 0.794i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.16 - 5.47i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.491 + 0.131i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 - 13.0iT - 61T^{2} \)
67 \( 1 + (-0.606 + 0.606i)T - 67iT^{2} \)
71 \( 1 + (3.01 + 11.2i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.377 - 1.40i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-5.80 - 10.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.23 - 1.23i)T + 83iT^{2} \)
89 \( 1 + (2.07 + 7.75i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-12.0 + 3.23i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.23720680538397990558855165839, −9.357182876136021203397121142991, −8.798752009366016374139295641433, −7.55940954078087830308817241069, −6.64560431722691885394705639988, −5.43985052404739122393345811993, −4.33794325269333978686184931063, −3.11605228259154587225221806545, −1.81005907832530771506450774136, −0.853369585422737951490649020746, 2.07859460856499437175242977033, 3.58846113226907401789462600710, 4.67366549291442224697015736442, 6.09338974405477120034395102348, 6.51737388632301146347093820282, 7.36110903920935795099587157476, 8.263627396574132815771619410897, 9.210771615107194717519524860067, 9.937693462613229526966872530580, 10.99069161543499956047224648196

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