Properties

Label 2-637-49.39-c1-0-33
Degree $2$
Conductor $637$
Sign $-0.603 + 0.797i$
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  + (−2.35 − 0.355i)2-s + (−1.45 + 0.993i)3-s + (3.51 + 1.08i)4-s + (0.0168 + 0.224i)5-s + (3.78 − 1.82i)6-s + (1.48 − 2.18i)7-s + (−3.59 − 1.73i)8-s + (0.0408 − 0.104i)9-s + (0.0401 − 0.535i)10-s + (−1.45 − 3.70i)11-s + (−6.19 + 1.91i)12-s + (−0.623 + 0.781i)13-s + (−4.27 + 4.62i)14-s + (−0.247 − 0.310i)15-s + (1.78 + 1.21i)16-s + (3.86 + 3.58i)17-s + ⋯
L(s)  = 1  + (−1.66 − 0.251i)2-s + (−0.841 + 0.573i)3-s + (1.75 + 0.541i)4-s + (0.00753 + 0.100i)5-s + (1.54 − 0.744i)6-s + (0.561 − 0.827i)7-s + (−1.27 − 0.612i)8-s + (0.0136 − 0.0347i)9-s + (0.0126 − 0.169i)10-s + (−0.438 − 1.11i)11-s + (−1.78 + 0.551i)12-s + (−0.172 + 0.216i)13-s + (−1.14 + 1.23i)14-s + (−0.0640 − 0.0802i)15-s + (0.445 + 0.304i)16-s + (0.937 + 0.869i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.603 + 0.797i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.603 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.603 + 0.797i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.603 + 0.797i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0869947 - 0.174859i\)
\(L(\frac12)\) \(\approx\) \(0.0869947 - 0.174859i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.48 + 2.18i)T \)
13 \( 1 + (0.623 - 0.781i)T \)
good2 \( 1 + (2.35 + 0.355i)T + (1.91 + 0.589i)T^{2} \)
3 \( 1 + (1.45 - 0.993i)T + (1.09 - 2.79i)T^{2} \)
5 \( 1 + (-0.0168 - 0.224i)T + (-4.94 + 0.745i)T^{2} \)
11 \( 1 + (1.45 + 3.70i)T + (-8.06 + 7.48i)T^{2} \)
17 \( 1 + (-3.86 - 3.58i)T + (1.27 + 16.9i)T^{2} \)
19 \( 1 + (2.13 + 3.69i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.75 - 5.33i)T + (1.71 - 22.9i)T^{2} \)
29 \( 1 + (0.546 - 2.39i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (2.58 - 4.48i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.94 + 1.21i)T + (30.5 - 20.8i)T^{2} \)
41 \( 1 + (10.2 + 4.93i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-9.48 + 4.56i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (8.80 + 1.32i)T + (44.9 + 13.8i)T^{2} \)
53 \( 1 + (7.43 + 2.29i)T + (43.7 + 29.8i)T^{2} \)
59 \( 1 + (-0.822 + 10.9i)T + (-58.3 - 8.79i)T^{2} \)
61 \( 1 + (13.9 - 4.28i)T + (50.4 - 34.3i)T^{2} \)
67 \( 1 + (-1.19 + 2.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.563 - 2.46i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-7.88 + 1.18i)T + (69.7 - 21.5i)T^{2} \)
79 \( 1 + (8.76 + 15.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.44 - 3.06i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-1.23 + 3.15i)T + (-65.2 - 60.5i)T^{2} \)
97 \( 1 + 5.13T + 97T^{2} \)
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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

−10.45965542379721078809082024442, −9.610869941292365789215426470223, −8.504581431263026261738969341307, −7.938878698604932325086367850829, −6.99730447481747836694046002762, −5.86586832938870464253616969893, −4.80608155542578891845489483353, −3.37704906067824481718694891488, −1.69621942818710551151603230820, −0.21765120187377977611000153573, 1.34099477626923965675164238040, 2.48432282869930735759694294370, 4.75727403185089511312750705917, 5.87348539176678140520431423261, 6.59245780228179764962403333383, 7.69146493617623459047902946100, 8.063567203679164251016743104117, 9.219392763329626816393666712565, 9.879623036774820645298711939836, 10.71347457655483978669236427924

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