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Properties

Label 2-637-7.4-c1-0-18
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  + (1 + 1.73i)2-s + (−0.999 + 1.73i)4-s + (1.5 + 2.59i)5-s + (1.5 + 2.59i)9-s + (−3 + 5.19i)10-s + (3 − 5.19i)11-s − 13-s + (1.99 + 3.46i)16-s + (−2 + 3.46i)17-s + (−3 + 5.19i)18-s + (−2.5 − 4.33i)19-s − 5.99·20-s + 12·22-s + (−1.5 − 2.59i)23-s + (−2 + 3.46i)25-s + (−1 − 1.73i)26-s + ⋯
L(s)  = 1  + (0.707 + 1.22i)2-s + (−0.499 + 0.866i)4-s + (0.670 + 1.16i)5-s + (0.5 + 0.866i)9-s + (−0.948 + 1.64i)10-s + (0.904 − 1.56i)11-s − 0.277·13-s + (0.499 + 0.866i)16-s + (−0.485 + 0.840i)17-s + (−0.707 + 1.22i)18-s + (−0.573 − 0.993i)19-s − 1.34·20-s + 2.55·22-s + (−0.312 − 0.541i)23-s + (−0.400 + 0.692i)25-s + (−0.196 − 0.339i)26-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: $\chi_{637} (508, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13318 + 2.28589i\)
\(L(\frac12)\) \(\approx\) \(1.13318 + 2.28589i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 1.73i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3 + 5.19i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.5 + 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 5T + 29T^{2} \)
31 \( 1 + (-1.5 + 2.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + (3.5 + 6.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4 - 6.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3 + 5.19i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (-6.5 + 11.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.5 + 2.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 15T + 83T^{2} \)
89 \( 1 + (1.5 + 2.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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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

−10.76930336283261111601906239084, −10.23108550068423304512400264697, −8.860086291941955628914762568412, −8.029994355851387376684716450214, −6.93909622768209686013821525932, −6.43600706997210459217068424160, −5.74503378456894661603662577883, −4.60456407011118138898494534144, −3.52097569193699672231567763651, −2.08139855061317724919995613635, 1.33070982566271400609252630348, 2.09201118382306068447341750076, 3.74241046224243632189636717717, 4.47414788048595035254945532500, 5.25236592027102490946166241079, 6.53987412117126131891073002678, 7.58478912016162589526448266690, 9.068090885679127142698830178716, 9.613822800953605735632754125364, 10.14210756010956150683716927382

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