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Properties

Label 2-637-13.4-c1-0-26
Degree $2$
Conductor $637$
Sign $0.711 - 0.702i$
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  + (1.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (0.5 + 0.866i)4-s − 1.73i·5-s + (−1.5 + 0.866i)6-s − 1.73i·8-s + (1 + 1.73i)9-s + (1.49 − 2.59i)10-s + (4.5 + 2.59i)11-s − 12-s + (1 − 3.46i)13-s + (1.49 + 0.866i)15-s + (2.49 − 4.33i)16-s + (3 + 5.19i)17-s + 3.46i·18-s + (−1.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (1.06 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (0.250 + 0.433i)4-s − 0.774i·5-s + (−0.612 + 0.353i)6-s − 0.612i·8-s + (0.333 + 0.577i)9-s + (0.474 − 0.821i)10-s + (1.35 + 0.783i)11-s − 0.288·12-s + (0.277 − 0.960i)13-s + (0.387 + 0.223i)15-s + (0.624 − 1.08i)16-s + (0.727 + 1.26i)17-s + 0.816i·18-s + (−0.344 + 0.198i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.711 - 0.702i$
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: \((2,\ 637,\ (\ :1/2),\ 0.711 - 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.33739 + 0.959412i\)
\(L(\frac12)\) \(\approx\) \(2.33739 + 0.959412i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 3.46i)T \)
good2 \( 1 + (-1.5 - 0.866i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 1.73iT - 5T^{2} \)
11 \( 1 + (-4.5 - 2.59i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.5 - 0.866i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.73iT - 31T^{2} \)
37 \( 1 + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.5 - 2.59i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.5 + 9.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 8.66iT - 47T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 + (3 - 1.73i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.5 + 4.33i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.5 + 0.866i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 8.66iT - 73T^{2} \)
79 \( 1 + 5T + 79T^{2} \)
83 \( 1 - 3.46iT - 83T^{2} \)
89 \( 1 + (6 + 3.46i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.5 + 2.59i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.54063427834550937794503280637, −9.955539897883459469536510071681, −8.950557914232390099311250175246, −7.905187421202558064518566865714, −6.87717086522113280013457682095, −5.88005622856155511040273888957, −5.17276796593610130225995289728, −4.31436506008169208408382198876, −3.64388388385592664360945142319, −1.44708519777574172275652910078, 1.40760246408540824846251294542, 2.91454216147683498369741216950, 3.72953607506592133535507672215, 4.69506539046643790491797726568, 6.09104426320387020544838645587, 6.55374477248910704439931241286, 7.57312751345662969479178831677, 8.895299466189142893195785011950, 9.666191730033022348124237170005, 11.06663851358317886124815166548

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