Properties

Label 2-637-13.4-c1-0-9
Degree $2$
Conductor $637$
Sign $-0.0651 - 0.997i$
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  + (0.900 + 0.520i)2-s + (−0.384 + 0.666i)3-s + (−0.459 − 0.795i)4-s + 1.67i·5-s + (−0.692 + 0.400i)6-s − 3.03i·8-s + (1.20 + 2.08i)9-s + (−0.871 + 1.51i)10-s + (−0.465 − 0.268i)11-s + 0.706·12-s + (1.96 + 3.02i)13-s + (−1.11 − 0.644i)15-s + (0.660 − 1.14i)16-s + (2.81 + 4.87i)17-s + 2.50i·18-s + (−1.74 + 1.00i)19-s + ⋯
L(s)  = 1  + (0.636 + 0.367i)2-s + (−0.222 + 0.384i)3-s + (−0.229 − 0.397i)4-s + 0.749i·5-s + (−0.282 + 0.163i)6-s − 1.07i·8-s + (0.401 + 0.695i)9-s + (−0.275 + 0.477i)10-s + (−0.140 − 0.0810i)11-s + 0.203·12-s + (0.544 + 0.838i)13-s + (−0.288 − 0.166i)15-s + (0.165 − 0.285i)16-s + (0.682 + 1.18i)17-s + 0.590i·18-s + (−0.400 + 0.231i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0651 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0651 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.0651 - 0.997i$
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: \((2,\ 637,\ (\ :1/2),\ -0.0651 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17539 + 1.25461i\)
\(L(\frac12)\) \(\approx\) \(1.17539 + 1.25461i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.96 - 3.02i)T \)
good2 \( 1 + (-0.900 - 0.520i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (0.384 - 0.666i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 1.67iT - 5T^{2} \)
11 \( 1 + (0.465 + 0.268i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.81 - 4.87i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.74 - 1.00i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.33 - 5.78i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.43 + 4.22i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.78iT - 31T^{2} \)
37 \( 1 + (1.19 + 0.690i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.07 - 0.620i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.63 + 2.82i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.79iT - 47T^{2} \)
53 \( 1 - 13.4T + 53T^{2} \)
59 \( 1 + (6.99 - 4.03i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.88 + 3.27i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.31 - 4.79i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.31 + 0.760i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 15.6iT - 73T^{2} \)
79 \( 1 - 8.26T + 79T^{2} \)
83 \( 1 + 9.42iT - 83T^{2} \)
89 \( 1 + (-1.85 - 1.06i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.58 + 3.22i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.44479611040411255482764702910, −10.33993832327579409236894702515, −9.213763215970023578733467910823, −8.048476423534061799371730088521, −7.00747892891902716978143057729, −6.15668039677912475794465560649, −5.40307824454573542963830954666, −4.30419606236942624144736087974, −3.56576029426236243286696806788, −1.74922207352107667982471717502, 0.851243326128917775690954206971, 2.61896245705565790724068709265, 3.75794866880827341455944934022, 4.73780869929701802417050671677, 5.56214460586333921331060263939, 6.70369682507822637268728912368, 7.81646345993751037773359311174, 8.559900312824160116313485103442, 9.416924814047334718751225741890, 10.50893300828491126483280005507

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