Properties

Label 2-637-49.8-c1-0-39
Degree $2$
Conductor $637$
Sign $0.971 + 0.236i$
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  + (0.933 + 1.17i)2-s + (−1.67 + 0.808i)3-s + (−0.0535 + 0.234i)4-s + (2.90 − 1.39i)5-s + (−2.51 − 1.21i)6-s + (−2.13 − 1.56i)7-s + (2.37 − 1.14i)8-s + (0.293 − 0.368i)9-s + (4.35 + 2.09i)10-s + (−2.84 − 3.56i)11-s + (−0.0997 − 0.436i)12-s + (0.623 + 0.781i)13-s + (−0.162 − 3.95i)14-s + (−3.74 + 4.69i)15-s + (3.98 + 1.91i)16-s + (−1.67 − 7.35i)17-s + ⋯
L(s)  = 1  + (0.659 + 0.827i)2-s + (−0.969 + 0.466i)3-s + (−0.0267 + 0.117i)4-s + (1.30 − 0.626i)5-s + (−1.02 − 0.494i)6-s + (−0.806 − 0.590i)7-s + (0.838 − 0.404i)8-s + (0.0979 − 0.122i)9-s + (1.37 + 0.662i)10-s + (−0.856 − 1.07i)11-s + (−0.0287 − 0.126i)12-s + (0.172 + 0.216i)13-s + (−0.0433 − 1.05i)14-s + (−0.967 + 1.21i)15-s + (0.996 + 0.479i)16-s + (−0.407 − 1.78i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.236i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 + 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.971 + 0.236i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (547, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.971 + 0.236i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66074 - 0.199036i\)
\(L(\frac12)\) \(\approx\) \(1.66074 - 0.199036i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.13 + 1.56i)T \)
13 \( 1 + (-0.623 - 0.781i)T \)
good2 \( 1 + (-0.933 - 1.17i)T + (-0.445 + 1.94i)T^{2} \)
3 \( 1 + (1.67 - 0.808i)T + (1.87 - 2.34i)T^{2} \)
5 \( 1 + (-2.90 + 1.39i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (2.84 + 3.56i)T + (-2.44 + 10.7i)T^{2} \)
17 \( 1 + (1.67 + 7.35i)T + (-15.3 + 7.37i)T^{2} \)
19 \( 1 - 4.53T + 19T^{2} \)
23 \( 1 + (-1.22 + 5.36i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (-1.76 - 7.71i)T + (-26.1 + 12.5i)T^{2} \)
31 \( 1 + 7.52T + 31T^{2} \)
37 \( 1 + (-1.96 - 8.61i)T + (-33.3 + 16.0i)T^{2} \)
41 \( 1 + (-4.09 + 1.97i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (-7.13 - 3.43i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (-0.663 - 0.831i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (2.34 - 10.2i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 + (3.16 + 1.52i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (1.26 + 5.56i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 - 5.37T + 67T^{2} \)
71 \( 1 + (-1.51 + 6.65i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-2.88 + 3.62i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 + 10.2T + 79T^{2} \)
83 \( 1 + (1.25 - 1.57i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (-8.37 + 10.5i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 + 0.268T + 97T^{2} \)
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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

−10.61778412626806992276842239830, −9.748190149953228939837849229857, −9.028392373221124979886778108697, −7.53625152631326343790142334323, −6.55048812898887139069186200195, −5.86562563460144722536341404235, −5.20461805926689023046819448464, −4.64963393184240850544320611159, −2.94231795585249877130071769702, −0.843303714241304073017402314067, 1.81219114651911382104373056608, 2.63013246097441834811197809326, 3.84698687889268686908242035294, 5.51377752333062838399332811954, 5.71657170936091764752753107136, 6.79849336596893284005279880477, 7.72055416827124506383537864843, 9.285947741242050560877646253613, 10.07485195360691148384441554634, 10.76322880269786353958152537270

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