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Properties

Label 2-637-13.10-c1-0-30
Degree $2$
Conductor $637$
Sign $0.932 + 0.360i$
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  + (−0.250 + 0.144i)2-s + (0.969 + 1.67i)3-s + (−0.958 + 1.65i)4-s − 4.29i·5-s + (−0.485 − 0.280i)6-s − 1.13i·8-s + (−0.381 + 0.660i)9-s + (0.620 + 1.07i)10-s + (1.60 − 0.928i)11-s − 3.71·12-s + (−2.29 − 2.77i)13-s + (7.21 − 4.16i)15-s + (−1.75 − 3.03i)16-s + (2.59 − 4.50i)17-s − 0.220i·18-s + (5.70 + 3.29i)19-s + ⋯
L(s)  = 1  + (−0.176 + 0.102i)2-s + (0.559 + 0.969i)3-s + (−0.479 + 0.829i)4-s − 1.92i·5-s + (−0.198 − 0.114i)6-s − 0.399i·8-s + (−0.127 + 0.220i)9-s + (0.196 + 0.339i)10-s + (0.484 − 0.279i)11-s − 1.07·12-s + (−0.637 − 0.770i)13-s + (1.86 − 1.07i)15-s + (−0.438 − 0.759i)16-s + (0.630 − 1.09i)17-s − 0.0519i·18-s + (1.30 + 0.756i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.360i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.932 + 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.932 + 0.360i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.932 + 0.360i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39280 - 0.260134i\)
\(L(\frac12)\) \(\approx\) \(1.39280 - 0.260134i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.29 + 2.77i)T \)
good2 \( 1 + (0.250 - 0.144i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-0.969 - 1.67i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 4.29iT - 5T^{2} \)
11 \( 1 + (-1.60 + 0.928i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.59 + 4.50i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.70 - 3.29i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.245 + 0.424i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.495 - 0.858i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.53iT - 31T^{2} \)
37 \( 1 + (2.75 - 1.59i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.129 + 0.0748i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.126 - 0.219i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 9.64iT - 47T^{2} \)
53 \( 1 - 0.777T + 53T^{2} \)
59 \( 1 + (-7.25 - 4.18i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.54 + 6.13i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.319 - 0.184i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.482 + 0.278i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 4.83iT - 73T^{2} \)
79 \( 1 + 7.82T + 79T^{2} \)
83 \( 1 + 6.38iT - 83T^{2} \)
89 \( 1 + (-8.26 + 4.77i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.30 - 2.48i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.834907512641509842005971434863, −9.625490229443680696091875352918, −8.882276996447023889313989803606, −8.157838574823148574970687103643, −7.46932631856806477959432400124, −5.56525950210506608597308153944, −4.81533100661801721664817224022, −4.02648260185274025658001599563, −3.09452092327241050190068031958, −0.849641603506794779388766327081, 1.60877142169044436049728050059, 2.57071692472602117211954418183, 3.76995509263519028963248180630, 5.31098006081719981290385627737, 6.55367943653702479697056710098, 6.98096535314849159284449923373, 7.81256518275558183958979201400, 8.946436868803687371139737375286, 9.946754538343055978530873855368, 10.43872895989441724060498645253

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