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Properties

Label 2-637-91.19-c1-0-6
Degree $2$
Conductor $637$
Sign $-0.747 - 0.664i$
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  + (−0.813 − 0.218i)2-s + 1.43i·3-s + (−1.11 − 0.645i)4-s + (−1.38 + 0.370i)5-s + (0.312 − 1.16i)6-s + (1.95 + 1.95i)8-s + 0.945·9-s + 1.20·10-s + (3.43 + 3.43i)11-s + (0.924 − 1.60i)12-s + (−1.04 − 3.44i)13-s + (−0.531 − 1.98i)15-s + (0.122 + 0.212i)16-s + (−1.49 + 2.58i)17-s + (−0.769 − 0.206i)18-s + (−4.44 − 4.44i)19-s + ⋯
L(s)  = 1  + (−0.575 − 0.154i)2-s + 0.827i·3-s + (−0.558 − 0.322i)4-s + (−0.618 + 0.165i)5-s + (0.127 − 0.476i)6-s + (0.692 + 0.692i)8-s + 0.315·9-s + 0.381·10-s + (1.03 + 1.03i)11-s + (0.266 − 0.462i)12-s + (−0.291 − 0.956i)13-s + (−0.137 − 0.512i)15-s + (0.0306 + 0.0531i)16-s + (−0.361 + 0.626i)17-s + (−0.181 − 0.0486i)18-s + (−1.01 − 1.01i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.747 - 0.664i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.747 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.747 - 0.664i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.747 - 0.664i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.183820 + 0.483218i\)
\(L(\frac12)\) \(\approx\) \(0.183820 + 0.483218i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.04 + 3.44i)T \)
good2 \( 1 + (0.813 + 0.218i)T + (1.73 + i)T^{2} \)
3 \( 1 - 1.43iT - 3T^{2} \)
5 \( 1 + (1.38 - 0.370i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-3.43 - 3.43i)T + 11iT^{2} \)
17 \( 1 + (1.49 - 2.58i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.44 + 4.44i)T + 19iT^{2} \)
23 \( 1 + (-1.02 + 0.590i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.77 - 4.81i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.75 - 6.54i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (1.44 - 5.40i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (4.71 - 1.26i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.90 - 1.67i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.51 - 5.63i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.89 + 5.01i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.88 + 10.7i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 - 9.18iT - 61T^{2} \)
67 \( 1 + (1.38 - 1.38i)T - 67iT^{2} \)
71 \( 1 + (3.19 + 0.855i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (0.482 + 0.129i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-3.47 + 6.02i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.22 + 3.22i)T + 83iT^{2} \)
89 \( 1 + (0.237 + 0.0636i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (2.43 - 9.08i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.61674137969858771955432606647, −10.08326345034472378914297417554, −9.235784379992449055508643270868, −8.622229856859108154311769458591, −7.53392365135395533146253242874, −6.60921943958142924182222733167, −5.04766531002927436120119872664, −4.48171447931970283284199246100, −3.50931448626180615604339886074, −1.62696612649371090108380668219, 0.36383057933548750501119586465, 1.83932614747450928878095424795, 3.81473872895435667495471586985, 4.33164144754367683594661358142, 6.01724731198755380675133162690, 6.94668580014313305332511271220, 7.67708246036136841638270603699, 8.456059333252759236177584460361, 9.165326664047737623361963042892, 10.00628965947567727232667097837

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