Properties

Label 2-637-13.3-c1-0-3
Degree $2$
Conductor $637$
Sign $-0.0128 + 0.999i$
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  + (−1.30 − 2.26i)2-s + (−1.30 − 2.26i)3-s + (−2.42 + 4.20i)4-s − 2.61·5-s + (−3.42 + 5.93i)6-s + 7.47·8-s + (−1.92 + 3.33i)9-s + (3.42 + 5.93i)10-s + (−0.927 − 1.60i)11-s + 12.7·12-s + (2.5 + 2.59i)13-s + (3.42 + 5.93i)15-s + (−4.92 − 8.53i)16-s + (−0.736 + 1.27i)17-s + 10.0·18-s + (−0.927 + 1.60i)19-s + ⋯
L(s)  = 1  + (−0.925 − 1.60i)2-s + (−0.755 − 1.30i)3-s + (−1.21 + 2.10i)4-s − 1.17·5-s + (−1.39 + 2.42i)6-s + 2.64·8-s + (−0.642 + 1.11i)9-s + (1.08 + 1.87i)10-s + (−0.279 − 0.484i)11-s + 3.66·12-s + (0.693 + 0.720i)13-s + (0.884 + 1.53i)15-s + (−1.23 − 2.13i)16-s + (−0.178 + 0.309i)17-s + 2.37·18-s + (−0.212 + 0.368i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0128 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.0128 + 0.999i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.0128 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.222055 - 0.224921i\)
\(L(\frac12)\) \(\approx\) \(0.222055 - 0.224921i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 2.59i)T \)
good2 \( 1 + (1.30 + 2.26i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.30 + 2.26i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 2.61T + 5T^{2} \)
11 \( 1 + (0.927 + 1.60i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.736 - 1.27i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.927 - 1.60i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.23 - 3.87i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.54 + 6.14i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.70T + 31T^{2} \)
37 \( 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.381 - 0.661i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.28 - 10.8i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 2.23T + 47T^{2} \)
53 \( 1 - 3.76T + 53T^{2} \)
59 \( 1 + (1.11 - 1.93i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3 - 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.35 - 11.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.09 + 12.2i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 6.70T + 83T^{2} \)
89 \( 1 + (-2.45 - 4.25i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.42 + 16.3i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.78368962930315983992221807256, −9.632412133044183266064694887320, −8.521184066760234686839209979709, −7.968531800087695512671194227769, −7.20069662318010363832121629460, −6.00362683498816012288218274278, −4.32562077038803765419804744541, −3.36740292400398492670489980159, −1.96236745724617875761502886578, −0.846359738284369259111786564214, 0.39171063169947395409479960353, 3.69928828572747184626824010892, 4.78766583298122437668728241726, 5.30026066457416476673587814141, 6.45082244553320685183094117946, 7.28142430154969307186149356697, 8.243514347111267237703203749680, 8.869650970196172709848997162113, 9.833657918126429242673495281099, 10.57928517715378383808720465396

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