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Properties

Label 2-637-91.74-c1-0-0
Degree $2$
Conductor $637$
Sign $0.0709 - 0.997i$
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  − 1.52·2-s + (−1.06 − 1.84i)3-s + 0.312·4-s + (0.294 + 0.510i)5-s + (1.61 + 2.80i)6-s + 2.56·8-s + (−0.760 + 1.31i)9-s + (−0.448 − 0.776i)10-s + (−0.760 − 1.31i)11-s + (−0.332 − 0.575i)12-s + (−3.32 + 1.39i)13-s + (0.626 − 1.08i)15-s − 4.52·16-s − 4.79·17-s + (1.15 − 2.00i)18-s + (0.841 − 1.45i)19-s + ⋯
L(s)  = 1  − 1.07·2-s + (−0.613 − 1.06i)3-s + 0.156·4-s + (0.131 + 0.228i)5-s + (0.660 + 1.14i)6-s + 0.907·8-s + (−0.253 + 0.439i)9-s + (−0.141 − 0.245i)10-s + (−0.229 − 0.397i)11-s + (−0.0959 − 0.166i)12-s + (−0.922 + 0.386i)13-s + (0.161 − 0.280i)15-s − 1.13·16-s − 1.16·17-s + (0.272 − 0.472i)18-s + (0.193 − 0.334i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0709 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0709 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.0709 - 0.997i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.0709 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.126055 + 0.117412i\)
\(L(\frac12)\) \(\approx\) \(0.126055 + 0.117412i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 + (3.32 - 1.39i)T \)
good2 \( 1 + 1.52T + 2T^{2} \)
3 \( 1 + (1.06 + 1.84i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.294 - 0.510i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.760 + 1.31i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 4.79T + 17T^{2} \)
19 \( 1 + (-0.841 + 1.45i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 1.77T + 23T^{2} \)
29 \( 1 + (3.44 - 5.96i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.04 - 5.27i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.40T + 37T^{2} \)
41 \( 1 + (-0.677 + 1.17i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.77 - 10.0i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.232 + 0.402i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.12 - 7.14i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 11.8T + 59T^{2} \)
61 \( 1 + (-1.24 + 2.14i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.78 - 6.55i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.30 + 5.71i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (8.18 - 14.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.48 - 12.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 + 16.4T + 89T^{2} \)
97 \( 1 + (0.486 + 0.843i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.90219205345461311756233214570, −9.801492635295605295822735402390, −9.028590442062745829632314052364, −8.188308625236039219580146503965, −7.06429022184783978374551425346, −6.88520196586742967211212927964, −5.53320963569313322871020250286, −4.41705888881218113834619753911, −2.51896304827374175188699923491, −1.23945567615879988125092159685, 0.15970903169895791234650601035, 2.12906343988599297098009574300, 4.01760582316073685561930398010, 4.82962943743118655429659318976, 5.61532630793213428661386036493, 7.10000731208091753548855675794, 7.85946762897607912178561571045, 8.999167034290977517081112168030, 9.567988370200848901800635621711, 10.19084549926169590493589723020

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