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Properties

Label 2-637-7.2-c1-0-10
Degree $2$
Conductor $637$
Sign $-0.605 + 0.795i$
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  + (−0.933 + 1.61i)2-s + (1.67 + 2.90i)3-s + (−0.741 − 1.28i)4-s + (−0.433 + 0.750i)5-s − 6.24·6-s − 0.965·8-s + (−4.10 + 7.11i)9-s + (−0.808 − 1.39i)10-s + (1.93 + 3.34i)11-s + (2.48 − 4.30i)12-s + 13-s − 2.90·15-s + (2.38 − 4.12i)16-s + (1.67 + 2.90i)17-s + (−7.66 − 13.2i)18-s + (2.69 − 4.66i)19-s + ⋯
L(s)  = 1  + (−0.659 + 1.14i)2-s + (0.966 + 1.67i)3-s + (−0.370 − 0.642i)4-s + (−0.193 + 0.335i)5-s − 2.55·6-s − 0.341·8-s + (−1.36 + 2.37i)9-s + (−0.255 − 0.442i)10-s + (0.582 + 1.00i)11-s + (0.716 − 1.24i)12-s + 0.277·13-s − 0.748·15-s + (0.595 − 1.03i)16-s + (0.406 + 0.703i)17-s + (−1.80 − 3.12i)18-s + (0.617 − 1.06i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.605 + 0.795i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.593492 - 1.19721i\)
\(L(\frac12)\) \(\approx\) \(0.593492 - 1.19721i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.933 - 1.61i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.67 - 2.90i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.433 - 0.750i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.93 - 3.34i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.67 - 2.90i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.69 + 4.66i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.62 + 4.54i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.69T + 29T^{2} \)
31 \( 1 + (3.78 + 6.55i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.41 + 4.18i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.06T + 41T^{2} \)
43 \( 1 - 4.03T + 43T^{2} \)
47 \( 1 + (-1.82 + 3.16i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.107 - 0.186i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.39 - 2.41i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.51 - 7.82i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.83 - 6.63i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.90T + 71T^{2} \)
73 \( 1 + (7.77 + 13.4i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.71 - 8.16i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.09T + 83T^{2} \)
89 \( 1 + (-0.209 + 0.362i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 7.11T + 97T^{2} \)
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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

−10.71835356600618835922273271249, −9.907553015569126528213349535960, −9.169121379783445561005655331658, −8.750255457015012957342084122698, −7.74278982051750506516985948995, −7.02777416599044518470485394884, −5.71582689338271742010158067868, −4.68944052630972722170433019945, −3.71405324026360066855495574597, −2.64590543205261566460566741738, 0.876548290787904101379704083952, 1.58387225089837139744746406110, 2.96550861823738999045185561556, 3.50262948687422162439013602052, 5.72534939361844554467738463343, 6.65225756346038314885482189277, 7.70445469248917101752760524612, 8.475884229959288557371536029933, 8.995400477218640636127098765262, 9.823320444566864810393589312746

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