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Properties

Label 2-637-91.54-c1-0-42
Degree $2$
Conductor $637$
Sign $-0.259 - 0.965i$
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  + (1.81 − 1.81i)2-s + (−2.33 − 1.34i)3-s − 4.59i·4-s + (−0.384 + 0.103i)5-s + (−6.68 + 1.79i)6-s + (−4.70 − 4.70i)8-s + (2.13 + 3.69i)9-s + (−0.511 + 0.885i)10-s + (−3.50 + 0.939i)11-s + (−6.19 + 10.7i)12-s + (−3.44 − 1.06i)13-s + (1.03 + 0.277i)15-s − 7.90·16-s + 4.09·17-s + (10.5 + 2.83i)18-s + (0.208 − 0.777i)19-s + ⋯
L(s)  = 1  + (1.28 − 1.28i)2-s + (−1.34 − 0.778i)3-s − 2.29i·4-s + (−0.172 + 0.0461i)5-s + (−2.72 + 0.731i)6-s + (−1.66 − 1.66i)8-s + (0.711 + 1.23i)9-s + (−0.161 + 0.280i)10-s + (−1.05 + 0.283i)11-s + (−1.78 + 3.09i)12-s + (−0.955 − 0.294i)13-s + (0.267 + 0.0717i)15-s − 1.97·16-s + 0.993·17-s + (2.49 + 0.668i)18-s + (0.0478 − 0.178i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.259 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.259 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.259 - 0.965i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.259 - 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.628473 + 0.819471i\)
\(L(\frac12)\) \(\approx\) \(0.628473 + 0.819471i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.44 + 1.06i)T \)
good2 \( 1 + (-1.81 + 1.81i)T - 2iT^{2} \)
3 \( 1 + (2.33 + 1.34i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.384 - 0.103i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (3.50 - 0.939i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 - 4.09T + 17T^{2} \)
19 \( 1 + (-0.208 + 0.777i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + 5.09iT - 23T^{2} \)
29 \( 1 + (-1.00 - 1.74i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.62 + 6.06i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.26 - 1.26i)T + 37iT^{2} \)
41 \( 1 + (0.578 - 2.15i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.65 + 1.53i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.19 + 8.19i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (4.54 + 7.87i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.25 - 5.25i)T - 59iT^{2} \)
61 \( 1 + (-2.40 + 1.38i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.30 - 4.85i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (0.582 + 2.17i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (4.78 + 1.28i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-1.80 + 3.13i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.36 + 5.36i)T + 83iT^{2} \)
89 \( 1 + (8.44 - 8.44i)T - 89iT^{2} \)
97 \( 1 + (-7.71 + 2.06i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.31765394335487527610345342688, −9.856106487200855387910945824921, −7.934078576232319615116775250114, −6.95534320323588955023808198378, −5.86264217268454989028207389283, −5.26588582392609198976846464155, −4.51286182393889239578912681072, −3.04472328968095687221298905693, −1.91878654295386374730328124696, −0.41943878477764337916530357249, 3.12234868435382194307173992787, 4.26705083395693226791788997864, 5.03726233963399603594493375752, 5.58987064535006077374492085024, 6.33619573164020275139612392084, 7.43537851564950064743716641315, 8.065414546713754187613601126486, 9.590360696549019472668693840606, 10.43756403293051497486635291231, 11.49776514720591299719776904002

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