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Properties

Label 2-637-7.4-c1-0-1
Degree $2$
Conductor $637$
Sign $0.386 - 0.922i$
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  + (−0.632 − 1.09i)2-s + (−1.31 + 2.27i)3-s + (0.199 − 0.344i)4-s + (−1.45 − 2.51i)5-s + 3.32·6-s − 3.03·8-s + (−1.95 − 3.37i)9-s + (−1.83 + 3.18i)10-s + (−1.01 + 1.76i)11-s + (0.523 + 0.906i)12-s − 13-s + 7.62·15-s + (1.52 + 2.63i)16-s + (1.99 − 3.46i)17-s + (−2.46 + 4.27i)18-s + (3.48 + 6.02i)19-s + ⋯
L(s)  = 1  + (−0.447 − 0.775i)2-s + (−0.758 + 1.31i)3-s + (0.0995 − 0.172i)4-s + (−0.649 − 1.12i)5-s + 1.35·6-s − 1.07·8-s + (−0.650 − 1.12i)9-s + (−0.580 + 1.00i)10-s + (−0.307 + 0.531i)11-s + (0.151 + 0.261i)12-s − 0.277·13-s + 1.96·15-s + (0.380 + 0.659i)16-s + (0.484 − 0.839i)17-s + (−0.582 + 1.00i)18-s + (0.798 + 1.38i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (508, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.363360 + 0.241700i\)
\(L(\frac12)\) \(\approx\) \(0.363360 + 0.241700i\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good2 \( 1 + (0.632 + 1.09i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.31 - 2.27i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.45 + 2.51i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.01 - 1.76i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.99 + 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.48 - 6.02i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.313 - 0.543i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.09T + 29T^{2} \)
31 \( 1 + (5.21 - 9.03i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.54 - 2.67i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.521T + 41T^{2} \)
43 \( 1 - 0.329T + 43T^{2} \)
47 \( 1 + (-5.27 - 9.13i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.55 - 6.16i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.01 + 1.76i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.20 - 2.07i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.34 - 12.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.60T + 71T^{2} \)
73 \( 1 + (-1.48 + 2.57i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.38 - 7.58i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12.8T + 83T^{2} \)
89 \( 1 + (1.34 + 2.32i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.32T + 97T^{2} \)
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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

−10.66860023664285565769151633152, −9.894148094285797605739541524837, −9.445936000234353015865173285834, −8.523629490695425173179604212382, −7.35474085463609778547411817706, −5.76706746378162283078596048519, −5.15039842121989766468154605313, −4.27604566745983872949076160342, −3.13454137785048425589551963484, −1.22425902318745276946291018345, 0.33497531996636792294717393949, 2.44075226651188237955903020029, 3.55191347602313058652062217936, 5.48056079535643673540215341020, 6.28855515985428871339851705578, 7.02324042732453490573856343570, 7.52885270322711963353762982316, 8.139344771287712465555593332749, 9.338717882485060035031226865536, 10.72972366373580767184668125085

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