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Properties

Label 2-637-91.33-c1-0-22
Degree $2$
Conductor $637$
Sign $0.999 - 0.0140i$
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  + (−0.629 + 2.35i)2-s − 1.97i·3-s + (−3.39 − 1.96i)4-s + (−0.0608 − 0.227i)5-s + (4.63 + 1.24i)6-s + (3.30 − 3.30i)8-s − 0.884·9-s + 0.572·10-s + (−2.02 + 2.02i)11-s + (−3.86 + 6.69i)12-s + (−2.03 + 2.97i)13-s + (−0.447 + 0.119i)15-s + (1.76 + 3.05i)16-s + (2.01 − 3.49i)17-s + (0.557 − 2.07i)18-s + (3.88 − 3.88i)19-s + ⋯
L(s)  = 1  + (−0.445 + 1.66i)2-s − 1.13i·3-s + (−1.69 − 0.980i)4-s + (−0.0272 − 0.101i)5-s + (1.89 + 0.506i)6-s + (1.16 − 1.16i)8-s − 0.294·9-s + 0.180·10-s + (−0.609 + 0.609i)11-s + (−1.11 + 1.93i)12-s + (−0.563 + 0.826i)13-s + (−0.115 + 0.0309i)15-s + (0.441 + 0.764i)16-s + (0.488 − 0.846i)17-s + (0.131 − 0.490i)18-s + (0.891 − 0.891i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0140i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0140i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.999 - 0.0140i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.999 - 0.0140i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.934515 + 0.00657313i\)
\(L(\frac12)\) \(\approx\) \(0.934515 + 0.00657313i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.03 - 2.97i)T \)
good2 \( 1 + (0.629 - 2.35i)T + (-1.73 - i)T^{2} \)
3 \( 1 + 1.97iT - 3T^{2} \)
5 \( 1 + (0.0608 + 0.227i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (2.02 - 2.02i)T - 11iT^{2} \)
17 \( 1 + (-2.01 + 3.49i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.88 + 3.88i)T - 19iT^{2} \)
23 \( 1 + (-5.23 + 3.02i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.54 + 6.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.22 + 0.595i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (10.0 + 2.68i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (0.0713 + 0.266i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-3.91 + 2.25i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.14 + 0.842i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (3.96 + 6.87i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.702 + 0.188i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + 2.28iT - 61T^{2} \)
67 \( 1 + (-1.88 - 1.88i)T + 67iT^{2} \)
71 \( 1 + (0.362 - 1.35i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (3.50 - 13.0i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (1.91 - 3.31i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.19 + 4.19i)T - 83iT^{2} \)
89 \( 1 + (-0.893 + 3.33i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-2.61 - 0.701i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.19341460707868682097329996105, −9.341960899092084872966769843721, −8.559833260866877156609430241364, −7.55802462516686010612722189754, −7.11274851046190891049864212299, −6.59464105574295063285976279372, −5.31200839838027245951560708388, −4.65196696347741137827563651775, −2.48420863968131875879917767145, −0.67801470821352899233399641159, 1.32594051721134550935589128163, 3.17977055461012548523285341489, 3.35690637415827601308468645211, 4.75959875837473600331432643456, 5.52669563903120644579374501209, 7.39266097103963823446392483340, 8.467121216328365271647322308642, 9.225665430307944048896290040075, 10.00320780826860671778338074510, 10.68808939719530730439792530209

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