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Properties

Label 2-637-49.44-c1-0-17
Degree $2$
Conductor $637$
Sign $-0.693 - 0.720i$
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  + (0.0482 − 0.00726i)2-s + (1.72 + 1.17i)3-s + (−1.90 + 0.588i)4-s + (−0.208 + 2.77i)5-s + (0.0917 + 0.0441i)6-s + (1.61 + 2.09i)7-s + (−0.175 + 0.0845i)8-s + (0.498 + 1.26i)9-s + (0.0101 + 0.135i)10-s + (−0.0497 + 0.126i)11-s + (−3.98 − 1.22i)12-s + (−0.623 − 0.781i)13-s + (0.0930 + 0.0893i)14-s + (−3.62 + 4.54i)15-s + (3.29 − 2.24i)16-s + (−0.921 + 0.855i)17-s + ⋯
L(s)  = 1  + (0.0340 − 0.00513i)2-s + (0.996 + 0.679i)3-s + (−0.954 + 0.294i)4-s + (−0.0930 + 1.24i)5-s + (0.0374 + 0.0180i)6-s + (0.609 + 0.792i)7-s + (−0.0620 + 0.0299i)8-s + (0.166 + 0.423i)9-s + (0.00320 + 0.0428i)10-s + (−0.0150 + 0.0382i)11-s + (−1.15 − 0.355i)12-s + (−0.172 − 0.216i)13-s + (0.0248 + 0.0238i)14-s + (−0.936 + 1.17i)15-s + (0.823 − 0.561i)16-s + (−0.223 + 0.207i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.693 - 0.720i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.693 - 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.693 - 0.720i$
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: \((2,\ 637,\ (\ :1/2),\ -0.693 - 0.720i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.613543 + 1.44298i\)
\(L(\frac12)\) \(\approx\) \(0.613543 + 1.44298i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.61 - 2.09i)T \)
13 \( 1 + (0.623 + 0.781i)T \)
good2 \( 1 + (-0.0482 + 0.00726i)T + (1.91 - 0.589i)T^{2} \)
3 \( 1 + (-1.72 - 1.17i)T + (1.09 + 2.79i)T^{2} \)
5 \( 1 + (0.208 - 2.77i)T + (-4.94 - 0.745i)T^{2} \)
11 \( 1 + (0.0497 - 0.126i)T + (-8.06 - 7.48i)T^{2} \)
17 \( 1 + (0.921 - 0.855i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (1.13 - 1.95i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.55 + 3.30i)T + (1.71 + 22.9i)T^{2} \)
29 \( 1 + (-0.352 - 1.54i)T + (-26.1 + 12.5i)T^{2} \)
31 \( 1 + (-3.15 - 5.47i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.96 + 0.915i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (3.94 - 1.89i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (2.74 + 1.32i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (-6.49 + 0.978i)T + (44.9 - 13.8i)T^{2} \)
53 \( 1 + (-4.29 + 1.32i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (-0.910 - 12.1i)T + (-58.3 + 8.79i)T^{2} \)
61 \( 1 + (-7.93 - 2.44i)T + (50.4 + 34.3i)T^{2} \)
67 \( 1 + (-0.797 - 1.38i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.64 + 11.6i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-12.3 - 1.85i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (0.0499 - 0.0865i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.75 + 7.21i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (1.17 + 3.00i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + 1.95T + 97T^{2} \)
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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

−10.51788586128696504945292014116, −10.08674717183256226313037733139, −8.972519980874159031083186717032, −8.502508316135458336412762648494, −7.71250290775883337814765314697, −6.45551598402853574019790620162, −5.22070946576408549341848156715, −4.15426942569352995736773516202, −3.29545804281958135120996369200, −2.39584373361607572183384294991, 0.798898283387836423886784039723, 1.99136445641883436057826953975, 3.76503176279843461609403158782, 4.61703010130304854059339345193, 5.41567213552008178849866421528, 6.93542851376210963609088967778, 8.076876001107789389002421428476, 8.314278854453291493791282348426, 9.221432852627181136468907333805, 9.923955264767753492654123063328

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