Properties

Label 2-637-91.16-c1-0-21
Degree $2$
Conductor $637$
Sign $0.748 + 0.662i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.748 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.748 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.748 + 0.662i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.748 + 0.662i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.955894 - 0.362335i\)
\(L(\frac12)\) \(\approx\) \(0.955894 - 0.362335i\)
\(L(\frac{3}{2})\) 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

−10.36736431185492229135331313793, −9.369127079388173119967960358301, −8.671719434771558849239211320386, −7.85175478368514212713736510970, −7.34864769611524238351124236749, −6.44903665323929366157216840905, −4.99245246217106583947010496839, −3.56162194254340414716475431990, −2.14002020018408487611894547576, −1.05844891543048948426270760138, 1.09219037361075807029991043589, 3.02409348741659261040715225342, 4.02640860745589707761275523583, 4.98291408467596015740749808579, 6.24063170058742796761330091418, 7.77084577824728841758040939310, 8.211019116710081452440121945098, 9.139573258107017555238963085759, 9.550668089363122082913597796574, 10.59321050818330418058534156553

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