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Properties

Label 2-637-49.44-c1-0-42
Degree $2$
Conductor $637$
Sign $0.998 - 0.0475i$
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  + (2.39 − 0.361i)2-s + (1.04 + 0.711i)3-s + (3.70 − 1.14i)4-s + (−0.165 + 2.20i)5-s + (2.75 + 1.32i)6-s + (1.24 − 2.33i)7-s + (4.10 − 1.97i)8-s + (−0.513 − 1.30i)9-s + (0.400 + 5.34i)10-s + (0.468 − 1.19i)11-s + (4.68 + 1.44i)12-s + (−0.623 − 0.781i)13-s + (2.13 − 6.04i)14-s + (−1.74 + 2.18i)15-s + (2.72 − 1.85i)16-s + (−4.26 + 3.95i)17-s + ⋯
L(s)  = 1  + (1.69 − 0.255i)2-s + (0.602 + 0.410i)3-s + (1.85 − 0.571i)4-s + (−0.0738 + 0.985i)5-s + (1.12 + 0.542i)6-s + (0.469 − 0.882i)7-s + (1.45 − 0.699i)8-s + (−0.171 − 0.436i)9-s + (0.126 + 1.69i)10-s + (0.141 − 0.359i)11-s + (1.35 + 0.416i)12-s + (−0.172 − 0.216i)13-s + (0.571 − 1.61i)14-s + (−0.449 + 0.563i)15-s + (0.681 − 0.464i)16-s + (−1.03 + 0.959i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.998 - 0.0475i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (534, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.998 - 0.0475i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.45560 + 0.105963i\)
\(L(\frac12)\) \(\approx\) \(4.45560 + 0.105963i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.24 + 2.33i)T \)
13 \( 1 + (0.623 + 0.781i)T \)
good2 \( 1 + (-2.39 + 0.361i)T + (1.91 - 0.589i)T^{2} \)
3 \( 1 + (-1.04 - 0.711i)T + (1.09 + 2.79i)T^{2} \)
5 \( 1 + (0.165 - 2.20i)T + (-4.94 - 0.745i)T^{2} \)
11 \( 1 + (-0.468 + 1.19i)T + (-8.06 - 7.48i)T^{2} \)
17 \( 1 + (4.26 - 3.95i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (2.67 - 4.63i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.46 + 3.21i)T + (1.71 + 22.9i)T^{2} \)
29 \( 1 + (-0.959 - 4.20i)T + (-26.1 + 12.5i)T^{2} \)
31 \( 1 + (-0.110 - 0.190i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.36 - 1.03i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (-6.42 + 3.09i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (2.67 + 1.28i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (1.92 - 0.290i)T + (44.9 - 13.8i)T^{2} \)
53 \( 1 + (4.96 - 1.53i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (0.301 + 4.02i)T + (-58.3 + 8.79i)T^{2} \)
61 \( 1 + (-1.13 - 0.350i)T + (50.4 + 34.3i)T^{2} \)
67 \( 1 + (3.67 + 6.36i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.613 + 2.68i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-1.06 - 0.160i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (-7.74 + 13.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.05 - 8.84i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (2.92 + 7.44i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 - 11.5T + 97T^{2} \)
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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

−10.69746401145105796627667757022, −10.27498835495957585621610395439, −8.786384121382147773658902713514, −7.76570275098823679537626252860, −6.55836705572849258804079345930, −6.14144497763998211268072952085, −4.65080078711866405840125635123, −3.88435684491746328118483739977, −3.26264592368040635529075321041, −2.09924631004209883379943602001, 2.05726864739434604501342651174, 2.76849728383629192135629816618, 4.42026503621609160984808122872, 4.83351715091833175122537745402, 5.77705009179139564740970090555, 6.84268369479886739379665777090, 7.78350735060146303344615863326, 8.681361000995141425087300087223, 9.428205445113457685043570457704, 11.22506758575822809789516407340

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