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Properties

Label 2-637-13.10-c1-0-12
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: $\chi_{637} (491, \cdot )$
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

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

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