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Properties

Label 2-637-13.4-c1-0-28
Degree $2$
Conductor $637$
Sign $0.921 + 0.388i$
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  + (1.10 + 0.638i)2-s + (−0.583 + 1.01i)3-s + (−0.185 − 0.320i)4-s − 1.81i·5-s + (−1.29 + 0.745i)6-s − 3.02i·8-s + (0.817 + 1.41i)9-s + (1.15 − 2.00i)10-s + (−2.40 − 1.38i)11-s + 0.432·12-s + (3.58 − 0.402i)13-s + (1.83 + 1.05i)15-s + (1.56 − 2.70i)16-s + (−1.37 − 2.37i)17-s + 2.08i·18-s + (5.08 − 2.93i)19-s + ⋯
L(s)  = 1  + (0.781 + 0.451i)2-s + (−0.337 + 0.583i)3-s + (−0.0925 − 0.160i)4-s − 0.811i·5-s + (−0.527 + 0.304i)6-s − 1.06i·8-s + (0.272 + 0.472i)9-s + (0.366 − 0.634i)10-s + (−0.725 − 0.418i)11-s + 0.124·12-s + (0.993 − 0.111i)13-s + (0.473 + 0.273i)15-s + (0.390 − 0.675i)16-s + (−0.332 − 0.576i)17-s + 0.492i·18-s + (1.16 − 0.673i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 + 0.388i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 + 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.921 + 0.388i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.921 + 0.388i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81647 - 0.367275i\)
\(L(\frac12)\) \(\approx\) \(1.81647 - 0.367275i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.58 + 0.402i)T \)
good2 \( 1 + (-1.10 - 0.638i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (0.583 - 1.01i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 1.81iT - 5T^{2} \)
11 \( 1 + (2.40 + 1.38i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.37 + 2.37i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.08 + 2.93i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.49 + 6.06i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.75 + 3.04i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.06iT - 31T^{2} \)
37 \( 1 + (-1.50 - 0.871i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.51 + 3.18i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.55 - 7.88i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 6.65iT - 47T^{2} \)
53 \( 1 + 10.4T + 53T^{2} \)
59 \( 1 + (2.66 - 1.53i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.540 - 0.936i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.34 - 2.50i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.35 + 1.35i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 7.67iT - 73T^{2} \)
79 \( 1 + 15.7T + 79T^{2} \)
83 \( 1 - 7.97iT - 83T^{2} \)
89 \( 1 + (-13.9 - 8.03i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-12.3 + 7.11i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.62044762590518839279276918451, −9.649543757418732552698777289820, −8.886825970077863287996359615603, −7.83791902379108236304348772825, −6.68148167889796386446204105626, −5.68440741199649013931755116313, −4.89379827455813777097649216142, −4.49250123848475734483028323202, −3.07508503180677753166180146724, −0.914360462783511827525090644663, 1.66303813185561268085416676053, 3.10718394128901578465300762119, 3.79450229206265301101799293091, 5.13593571017413300746496658797, 6.02893518718146194320923604028, 7.04529947149689818335969088401, 7.78279721911240896529314990356, 8.866878060729724003729231388061, 10.00830093363090494994050177698, 10.97911470107977134484234959049

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