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Properties

Label 2-637-91.24-c1-0-34
Degree $2$
Conductor $637$
Sign $-0.646 + 0.762i$
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  + (0.263 − 0.0707i)2-s − 2.50i·3-s + (−1.66 + 0.962i)4-s + (1.43 + 0.383i)5-s + (−0.177 − 0.661i)6-s + (−0.758 + 0.758i)8-s − 3.28·9-s + 0.404·10-s + (3.24 − 3.24i)11-s + (2.41 + 4.17i)12-s + (−2.80 − 2.26i)13-s + (0.960 − 3.58i)15-s + (1.77 − 3.08i)16-s + (−1.17 − 2.02i)17-s + (−0.866 + 0.232i)18-s + (1.19 − 1.19i)19-s + ⋯
L(s)  = 1  + (0.186 − 0.0500i)2-s − 1.44i·3-s + (−0.833 + 0.481i)4-s + (0.639 + 0.171i)5-s + (−0.0723 − 0.270i)6-s + (−0.268 + 0.268i)8-s − 1.09·9-s + 0.127·10-s + (0.978 − 0.978i)11-s + (0.696 + 1.20i)12-s + (−0.778 − 0.627i)13-s + (0.247 − 0.925i)15-s + (0.444 − 0.770i)16-s + (−0.283 − 0.491i)17-s + (−0.204 + 0.0547i)18-s + (0.274 − 0.274i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.646 + 0.762i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.646 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.646 + 0.762i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (570, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.646 + 0.762i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.525084 - 1.13309i\)
\(L(\frac12)\) \(\approx\) \(0.525084 - 1.13309i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.80 + 2.26i)T \)
good2 \( 1 + (-0.263 + 0.0707i)T + (1.73 - i)T^{2} \)
3 \( 1 + 2.50iT - 3T^{2} \)
5 \( 1 + (-1.43 - 0.383i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-3.24 + 3.24i)T - 11iT^{2} \)
17 \( 1 + (1.17 + 2.02i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.19 + 1.19i)T - 19iT^{2} \)
23 \( 1 + (4.15 + 2.39i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.87 + 4.98i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.52 - 5.69i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-2.20 - 8.24i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (0.829 + 0.222i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-1.70 - 0.981i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.07 - 7.75i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-6.54 + 11.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.400 + 1.49i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + 9.76iT - 61T^{2} \)
67 \( 1 + (0.385 + 0.385i)T + 67iT^{2} \)
71 \( 1 + (-1.77 + 0.474i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-2.28 + 0.611i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (2.13 + 3.69i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.88 + 3.88i)T - 83iT^{2} \)
89 \( 1 + (-13.5 + 3.63i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-0.734 - 2.73i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.11201803725903565659809468177, −9.294789864055326372022959836646, −8.320781020801882904699401480424, −7.72449524502382062514379389054, −6.60883331450802951118662211193, −5.94825185295215623977855693422, −4.78038233653529541822008209086, −3.36225460183386113504940300965, −2.24630149769216808343975453061, −0.66176721819447072435798475140, 1.87656554459809877863350395000, 3.87390841696006021027548686950, 4.27969437335308876602999323196, 5.27514320110362971812394259919, 5.99467600766276282959270455196, 7.32599957297447501404839789942, 8.846510179402290940317680050448, 9.411677936989349545787884316208, 9.819445433373003313620969656310, 10.50278253953511388010349887546

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