Properties

Label 2-637-91.74-c1-0-14
Degree $2$
Conductor $637$
Sign $0.355 + 0.934i$
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  − 0.579·2-s + (−0.946 − 1.63i)3-s − 1.66·4-s + (0.736 + 1.27i)5-s + (0.548 + 0.950i)6-s + 2.12·8-s + (−0.289 + 0.502i)9-s + (−0.427 − 0.739i)10-s + (−0.289 − 0.502i)11-s + (1.57 + 2.72i)12-s + (0.128 + 3.60i)13-s + (1.39 − 2.41i)15-s + 2.09·16-s + 1.19·17-s + (0.168 − 0.291i)18-s + (−0.230 + 0.399i)19-s + ⋯
L(s)  = 1  − 0.409·2-s + (−0.546 − 0.946i)3-s − 0.831·4-s + (0.329 + 0.570i)5-s + (0.223 + 0.387i)6-s + 0.751·8-s + (−0.0966 + 0.167i)9-s + (−0.135 − 0.233i)10-s + (−0.0874 − 0.151i)11-s + (0.454 + 0.787i)12-s + (0.0357 + 0.999i)13-s + (0.359 − 0.623i)15-s + 0.523·16-s + 0.290·17-s + (0.0396 − 0.0686i)18-s + (−0.0528 + 0.0915i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.355 + 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.355 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.355 + 0.934i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.355 + 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.673760 - 0.464427i\)
\(L(\frac12)\) \(\approx\) \(0.673760 - 0.464427i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.128 - 3.60i)T \)
good2 \( 1 + 0.579T + 2T^{2} \)
3 \( 1 + (0.946 + 1.63i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.736 - 1.27i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.289 + 0.502i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 1.19T + 17T^{2} \)
19 \( 1 + (0.230 - 0.399i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 2.36T + 23T^{2} \)
29 \( 1 + (-3.44 + 5.96i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.22 + 3.84i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 9.16T + 37T^{2} \)
41 \( 1 + (2.00 - 3.47i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.02 + 6.97i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.75 + 9.97i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.69 + 8.13i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 0.240T + 59T^{2} \)
61 \( 1 + (-3.86 + 6.69i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.724 - 1.25i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.25 - 10.8i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.84 - 3.19i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.03 + 13.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 15.4T + 83T^{2} \)
89 \( 1 + 2.49T + 89T^{2} \)
97 \( 1 + (-7.82 - 13.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.21486810813399703292187015339, −9.669204887449777328942757721623, −8.609825280858296438312486997968, −7.77347675637450382920699975295, −6.78770393394062905603242249766, −6.16782541660733044621504550857, −4.98287107339678412136637646125, −3.81681242304431167174811833702, −2.14701931989856412948559058917, −0.72273975159116333587483893882, 1.11492703540708945793058665478, 3.24683351912744735327667771579, 4.59843725726093570780371400235, 5.01187757739884982110026878344, 5.91076266754457141825335745633, 7.44530904655738560712282027159, 8.371983832712743482142336303237, 9.184507679584010355832707849004, 9.896128757694921097189792361247, 10.49550404578259339978565006478

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