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Properties

Label 2-637-91.4-c1-0-19
Degree $2$
Conductor $637$
Sign $0.701 - 0.712i$
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  + 0.499i·2-s + (0.424 + 0.735i)3-s + 1.75·4-s + (0.902 − 0.521i)5-s + (−0.367 + 0.212i)6-s + 1.87i·8-s + (1.13 − 1.97i)9-s + (0.260 + 0.451i)10-s + (−3.43 + 1.98i)11-s + (0.743 + 1.28i)12-s + (3.57 − 0.468i)13-s + (0.767 + 0.442i)15-s + 2.56·16-s − 0.142·17-s + (0.986 + 0.569i)18-s + (4.77 + 2.75i)19-s + ⋯
L(s)  = 1  + 0.353i·2-s + (0.245 + 0.424i)3-s + 0.875·4-s + (0.403 − 0.233i)5-s + (−0.150 + 0.0867i)6-s + 0.662i·8-s + (0.379 − 0.657i)9-s + (0.0824 + 0.142i)10-s + (−1.03 + 0.598i)11-s + (0.214 + 0.371i)12-s + (0.991 − 0.129i)13-s + (0.198 + 0.114i)15-s + 0.640·16-s − 0.0344·17-s + (0.232 + 0.134i)18-s + (1.09 + 0.632i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.701 - 0.712i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.701 - 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.00469 + 0.839745i\)
\(L(\frac12)\) \(\approx\) \(2.00469 + 0.839745i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.57 + 0.468i)T \)
good2 \( 1 - 0.499iT - 2T^{2} \)
3 \( 1 + (-0.424 - 0.735i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.902 + 0.521i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.43 - 1.98i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 0.142T + 17T^{2} \)
19 \( 1 + (-4.77 - 2.75i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 4.39T + 23T^{2} \)
29 \( 1 + (-4.19 + 7.27i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.46 + 1.42i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.843iT - 37T^{2} \)
41 \( 1 + (10.4 + 6.04i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.41 - 4.17i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.94 + 2.27i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.139 + 0.242i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 10.7iT - 59T^{2} \)
61 \( 1 + (2.93 - 5.07i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.45 - 2.57i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.20 - 1.84i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.72 + 3.30i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.96 + 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.87iT - 83T^{2} \)
89 \( 1 + 1.74iT - 89T^{2} \)
97 \( 1 + (2.34 - 1.35i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.35743052725265615190877453874, −10.04641949338416457424804879935, −8.939930148376424857794804043451, −7.944073843003080557372664554054, −7.22544459974293462734439892719, −6.08401658940197472286193812622, −5.46927724900036840943175614285, −4.08787246961171320113304647026, −2.94582062405899596132904619327, −1.62149544095410532181072251436, 1.42224929190624269986706684322, 2.52295248866495828648052508882, 3.41547353849683713012757061836, 5.05410081288077447009832954976, 6.09411939394570285326383491529, 6.94104066470470107199477140269, 7.79971208768693226469328170667, 8.571767848268036105942937957080, 9.922673817112999315060781941328, 10.56130778446612340777040116202

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