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Properties

Label 2-637-49.39-c1-0-23
Degree $2$
Conductor $637$
Sign $0.993 + 0.112i$
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  + (−2.41 − 0.363i)2-s + (1.79 − 1.22i)3-s + (3.76 + 1.16i)4-s + (−0.0804 − 1.07i)5-s + (−4.76 + 2.29i)6-s + (1.16 + 2.37i)7-s + (−4.26 − 2.05i)8-s + (0.622 − 1.58i)9-s + (−0.196 + 2.61i)10-s + (0.766 + 1.95i)11-s + (8.17 − 2.52i)12-s + (−0.623 + 0.781i)13-s + (−1.94 − 6.14i)14-s + (−1.45 − 1.82i)15-s + (3.02 + 2.06i)16-s + (3.11 + 2.89i)17-s + ⋯
L(s)  = 1  + (−1.70 − 0.256i)2-s + (1.03 − 0.705i)3-s + (1.88 + 0.581i)4-s + (−0.0359 − 0.479i)5-s + (−1.94 + 0.936i)6-s + (0.440 + 0.897i)7-s + (−1.50 − 0.726i)8-s + (0.207 − 0.528i)9-s + (−0.0619 + 0.827i)10-s + (0.231 + 0.588i)11-s + (2.35 − 0.727i)12-s + (−0.172 + 0.216i)13-s + (−0.520 − 1.64i)14-s + (−0.375 − 0.471i)15-s + (0.757 + 0.516i)16-s + (0.756 + 0.701i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.112i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.993 + 0.112i$
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: \((2,\ 637,\ (\ :1/2),\ 0.993 + 0.112i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00332 - 0.0568003i\)
\(L(\frac12)\) \(\approx\) \(1.00332 - 0.0568003i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.16 - 2.37i)T \)
13 \( 1 + (0.623 - 0.781i)T \)
good2 \( 1 + (2.41 + 0.363i)T + (1.91 + 0.589i)T^{2} \)
3 \( 1 + (-1.79 + 1.22i)T + (1.09 - 2.79i)T^{2} \)
5 \( 1 + (0.0804 + 1.07i)T + (-4.94 + 0.745i)T^{2} \)
11 \( 1 + (-0.766 - 1.95i)T + (-8.06 + 7.48i)T^{2} \)
17 \( 1 + (-3.11 - 2.89i)T + (1.27 + 16.9i)T^{2} \)
19 \( 1 + (-3.33 - 5.77i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.214 - 0.198i)T + (1.71 - 22.9i)T^{2} \)
29 \( 1 + (-2.01 + 8.81i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (4.57 - 7.91i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.01 - 0.930i)T + (30.5 - 20.8i)T^{2} \)
41 \( 1 + (5.95 + 2.86i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (1.53 - 0.737i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-1.89 - 0.285i)T + (44.9 + 13.8i)T^{2} \)
53 \( 1 + (7.75 + 2.39i)T + (43.7 + 29.8i)T^{2} \)
59 \( 1 + (-0.368 + 4.91i)T + (-58.3 - 8.79i)T^{2} \)
61 \( 1 + (-14.4 + 4.45i)T + (50.4 - 34.3i)T^{2} \)
67 \( 1 + (-3.46 + 5.99i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.297 + 1.30i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-9.67 + 1.45i)T + (69.7 - 21.5i)T^{2} \)
79 \( 1 + (-0.363 - 0.628i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.0925 - 0.116i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-2.32 + 5.91i)T + (-65.2 - 60.5i)T^{2} \)
97 \( 1 + 6.91T + 97T^{2} \)
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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

−10.15344958410051277138538737835, −9.515682302224945191859740766618, −8.636372401943183904479748052538, −8.217549065285724786629887614928, −7.59586683102845104848461976548, −6.59963636461420214861331598213, −5.19671251485391561357513582300, −3.33773079106767245285146592486, −2.07235231254932005883996587740, −1.46265860029448572311790962119, 0.948402678495125610437284536313, 2.66286473800849511768990812549, 3.60526725668823392219784037648, 5.10928615675491865776655576918, 6.75851502233524435243883325404, 7.35186411147363972065634941682, 8.170546932388462894484487250986, 8.945691005590163856601020436879, 9.558755684159346377138320470426, 10.30628935998419267272276690176

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