Properties

Label 2-637-13.10-c1-0-17
Degree $2$
Conductor $637$
Sign $0.252 - 0.967i$
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 + (1 + 1.73i)3-s + (0.5 − 0.866i)4-s − 1.73i·5-s + (−3 − 1.73i)6-s − 1.73i·8-s + (−0.499 + 0.866i)9-s + (1.49 + 2.59i)10-s + 2·12-s + (2.5 − 2.59i)13-s + (2.99 − 1.73i)15-s + (2.49 + 4.33i)16-s + (−1.5 + 2.59i)17-s − 1.73i·18-s + (3 + 1.73i)19-s + (−1.50 − 0.866i)20-s + ⋯
L(s)  = 1  + (−1.06 + 0.612i)2-s + (0.577 + 0.999i)3-s + (0.250 − 0.433i)4-s − 0.774i·5-s + (−1.22 − 0.707i)6-s − 0.612i·8-s + (−0.166 + 0.288i)9-s + (0.474 + 0.821i)10-s + 0.577·12-s + (0.693 − 0.720i)13-s + (0.774 − 0.447i)15-s + (0.624 + 1.08i)16-s + (−0.363 + 0.630i)17-s − 0.408i·18-s + (0.688 + 0.397i)19-s + (−0.335 − 0.193i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.252 - 0.967i$
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: \((2,\ 637,\ (\ :1/2),\ 0.252 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.840982 + 0.649602i\)
\(L(\frac12)\) \(\approx\) \(0.840982 + 0.649602i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 + 2.59i)T \)
good2 \( 1 + (1.5 - 0.866i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-1 - 1.73i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 1.73iT - 5T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3 - 1.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 + (-7.5 + 4.33i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.5 + 2.59i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.46iT - 47T^{2} \)
53 \( 1 + 3T + 53T^{2} \)
59 \( 1 + (6 + 3.46i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3 + 1.73i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3 - 1.73i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 1.73iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 13.8iT - 83T^{2} \)
89 \( 1 + (-6 + 3.46i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (6 + 3.46i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.35634903222647307214249563420, −9.523042953568066935061081759641, −9.093346863751488825405663314713, −8.312282453542354675124519549229, −7.67708443227877586900418878810, −6.42761798679709011725480848724, −5.31126613198605125588421274489, −4.12940735671574564333337824063, −3.28612597031454256910109698669, −1.12344547484835562828742641585, 1.05120114925961720351497648493, 2.27433639748080715438872033636, 3.03066183000832866525146587183, 4.78070743596815260235133973597, 6.32759656866695939344260748572, 7.12105337510844211703819790809, 7.87655938772660249829676498002, 8.784606032870381705088067350927, 9.368023433848938627261473378530, 10.45077125195733125310922020347

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