Properties

Label 2-637-91.4-c1-0-29
Degree $2$
Conductor $637$
Sign $-0.923 + 0.384i$
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  − 1.27i·2-s + (−0.583 − 1.01i)3-s + 0.370·4-s + (−1.57 + 0.907i)5-s + (−1.29 + 0.745i)6-s − 3.02i·8-s + (0.817 − 1.41i)9-s + (1.15 + 2.00i)10-s + (2.40 − 1.38i)11-s + (−0.216 − 0.374i)12-s + (3.58 − 0.402i)13-s + (1.83 + 1.05i)15-s − 3.12·16-s + 2.74·17-s + (−1.80 − 1.04i)18-s + (−5.08 − 2.93i)19-s + ⋯
L(s)  = 1  − 0.902i·2-s + (−0.337 − 0.583i)3-s + 0.185·4-s + (−0.702 + 0.405i)5-s + (−0.527 + 0.304i)6-s − 1.06i·8-s + (0.272 − 0.472i)9-s + (0.366 + 0.634i)10-s + (0.725 − 0.418i)11-s + (−0.0624 − 0.108i)12-s + (0.993 − 0.111i)13-s + (0.473 + 0.273i)15-s − 0.780·16-s + 0.665·17-s + (−0.426 − 0.246i)18-s + (−1.16 − 0.673i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.384i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 + 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.923 + 0.384i$
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: \((2,\ 637,\ (\ :1/2),\ -0.923 + 0.384i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.252077 - 1.26071i\)
\(L(\frac12)\) \(\approx\) \(0.252077 - 1.26071i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.58 + 0.402i)T \)
good2 \( 1 + 1.27iT - 2T^{2} \)
3 \( 1 + (0.583 + 1.01i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.57 - 0.907i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.40 + 1.38i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 2.74T + 17T^{2} \)
19 \( 1 + (5.08 + 2.93i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 6.99T + 23T^{2} \)
29 \( 1 + (-1.75 + 3.04i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.79 + 1.03i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 1.74iT - 37T^{2} \)
41 \( 1 + (5.51 + 3.18i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.55 - 7.88i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.76 + 3.32i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.24 + 9.08i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 3.07iT - 59T^{2} \)
61 \( 1 + (-0.540 + 0.936i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.34 - 2.50i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.35 + 1.35i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.64 - 3.83i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.86 - 13.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.97iT - 83T^{2} \)
89 \( 1 + 16.0iT - 89T^{2} \)
97 \( 1 + (-12.3 + 7.11i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.46825590203539907769822152079, −9.571165301691853602903909469256, −8.462333976981500583302919441003, −7.43665299025909111235885632273, −6.57351112823867322724584107200, −5.98413851103102059266273010787, −4.00328618217459018221643632827, −3.55191741944683488981942877706, −2.04473576669002256554009186420, −0.74654564890158799532617812296, 1.85373254220572528227519880723, 3.81329651604730884510982313762, 4.52590097769786824750339231136, 5.68410459753261740800968538790, 6.39596787488319186585940143294, 7.50646935550833828931689104014, 8.174654111036473461785807778020, 8.963858678057750085071769265745, 10.27205735268847772830881384449, 10.80559714721814563923901429153

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