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Properties

Label 2-637-91.89-c1-0-6
Degree $2$
Conductor $637$
Sign $0.656 - 0.754i$
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  + (−1.65 − 1.65i)2-s + (2.39 + 1.38i)3-s + 3.50i·4-s + (0.412 + 1.53i)5-s + (−1.67 − 6.25i)6-s + (2.50 − 2.50i)8-s + (2.30 + 4.00i)9-s + (1.87 − 3.23i)10-s + (0.813 + 3.03i)11-s + (−4.84 + 8.39i)12-s + (−1.04 + 3.44i)13-s + (−1.13 + 4.24i)15-s − 1.29·16-s − 0.641·17-s + (2.80 − 10.4i)18-s + (−7.61 − 2.04i)19-s + ⋯
L(s)  = 1  + (−1.17 − 1.17i)2-s + (1.38 + 0.796i)3-s + 1.75i·4-s + (0.184 + 0.688i)5-s + (−0.684 − 2.55i)6-s + (0.886 − 0.886i)8-s + (0.769 + 1.33i)9-s + (0.591 − 1.02i)10-s + (0.245 + 0.914i)11-s + (−1.39 + 2.42i)12-s + (−0.290 + 0.956i)13-s + (−0.293 + 1.09i)15-s − 0.324·16-s − 0.155·17-s + (0.661 − 2.46i)18-s + (−1.74 − 0.468i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 - 0.754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.656 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.656 - 0.754i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (362, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.656 - 0.754i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06748 + 0.485816i\)
\(L(\frac12)\) \(\approx\) \(1.06748 + 0.485816i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.04 - 3.44i)T \)
good2 \( 1 + (1.65 + 1.65i)T + 2iT^{2} \)
3 \( 1 + (-2.39 - 1.38i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.412 - 1.53i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.813 - 3.03i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + 0.641T + 17T^{2} \)
19 \( 1 + (7.61 + 2.04i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + 0.146iT - 23T^{2} \)
29 \( 1 + (-1.49 - 2.58i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (6.46 + 1.73i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (2.75 - 2.75i)T - 37iT^{2} \)
41 \( 1 + (-5.60 - 1.50i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-2.42 - 1.40i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.04 + 0.816i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-3.66 - 6.34i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.93 - 2.93i)T + 59iT^{2} \)
61 \( 1 + (-3.90 + 2.25i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.36 + 0.366i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-13.8 + 3.70i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-1.82 + 6.81i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-0.316 + 0.548i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.07 + 1.07i)T - 83iT^{2} \)
89 \( 1 + (-9.60 - 9.60i)T + 89iT^{2} \)
97 \( 1 + (-0.0487 - 0.181i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.53843828027842234550071313236, −9.743102923131268584450295558440, −9.127810675323766776607779435260, −8.627411915901564087291878295008, −7.59509725203975571305421844126, −6.66240333631202600973021184473, −4.53504848999822076065552379161, −3.71221062940833013415930034045, −2.53324411341315071115373377573, −2.03722659257642604236417171096, 0.791930117588839082651286808422, 2.16831259424141905090545001325, 3.67050300587913366703867229306, 5.40737304091804598297012333972, 6.34730016209952083550394902778, 7.25835670485000188833687159479, 8.026458154849181949227685211052, 8.680603767934134280479755799879, 8.939903035050055453645097359060, 9.985922537179438384596310216768

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