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Properties

Label 1-637-637.537-r1-0-0
Degree $1$
Conductor $637$
Sign $0.995 - 0.0938i$
Analytic cond. $68.4551$
Root an. cond. $68.4551$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0747 + 0.997i)2-s + (0.222 − 0.974i)3-s + (−0.988 − 0.149i)4-s + (0.955 − 0.294i)5-s + (0.955 + 0.294i)6-s + (0.222 − 0.974i)8-s + (−0.900 − 0.433i)9-s + (0.222 + 0.974i)10-s + (0.900 − 0.433i)11-s + (−0.365 + 0.930i)12-s + (−0.0747 − 0.997i)15-s + (0.955 + 0.294i)16-s + (−0.365 + 0.930i)17-s + (0.5 − 0.866i)18-s + 19-s + (−0.988 + 0.149i)20-s + ⋯
L(s)  = 1  + (−0.0747 + 0.997i)2-s + (0.222 − 0.974i)3-s + (−0.988 − 0.149i)4-s + (0.955 − 0.294i)5-s + (0.955 + 0.294i)6-s + (0.222 − 0.974i)8-s + (−0.900 − 0.433i)9-s + (0.222 + 0.974i)10-s + (0.900 − 0.433i)11-s + (−0.365 + 0.930i)12-s + (−0.0747 − 0.997i)15-s + (0.955 + 0.294i)16-s + (−0.365 + 0.930i)17-s + (0.5 − 0.866i)18-s + 19-s + (−0.988 + 0.149i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.995 - 0.0938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.995 - 0.0938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.995 - 0.0938i$
Analytic conductor: \(68.4551\)
Root analytic conductor: \(68.4551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (537, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 637,\ (1:\ ),\ 0.995 - 0.0938i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.585608525 - 0.1216562000i\)
\(L(\frac12)\) \(\approx\) \(2.585608525 - 0.1216562000i\)
\(L(1)\) \(\approx\) \(1.318377274 + 0.09092711213i\)
\(L(1)\) \(\approx\) \(1.318377274 + 0.09092711213i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.0747 + 0.997i)T \)
3 \( 1 + (0.222 - 0.974i)T \)
5 \( 1 + (0.955 - 0.294i)T \)
11 \( 1 + (0.900 - 0.433i)T \)
17 \( 1 + (-0.365 + 0.930i)T \)
19 \( 1 + T \)
23 \( 1 + (0.365 + 0.930i)T \)
29 \( 1 + (0.365 - 0.930i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.988 - 0.149i)T \)
41 \( 1 + (0.955 - 0.294i)T \)
43 \( 1 + (0.955 + 0.294i)T \)
47 \( 1 + (0.826 + 0.563i)T \)
53 \( 1 + (0.365 + 0.930i)T \)
59 \( 1 + (-0.733 + 0.680i)T \)
61 \( 1 + (-0.623 - 0.781i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.365 - 0.930i)T \)
73 \( 1 + (0.826 - 0.563i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.900 - 0.433i)T \)
89 \( 1 + (0.826 - 0.563i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.37450743765387660944419832019, −21.929773774630392595378070944386, −21.01644879055552457291091740434, −20.35789003029322446434997330082, −19.80920766185160996262039550035, −18.58998525121271612894546388895, −17.89552537581862411303095363549, −17.03903183677515353215394727111, −16.28862920306044225605323447306, −14.90995725580100831880289924056, −14.26423173371199350033193131889, −13.64458555884086964074528186291, −12.55894165805445754455962916332, −11.47854268009073768306337930489, −10.82589026889115105509581165263, −9.85214379866939003315160502695, −9.38494414642690223882047183151, −8.706236969247504814233004855117, −7.255517102897668153825711209402, −5.86836373119633451754324628067, −4.93958348245301776811847018709, −4.10721270775756021146805496956, −2.982563700310604966706912440444, −2.28341108765582067621019730896, −0.93255697234165115689477385615, 0.826969760795615970742042643953, 1.609620236903251073559259433945, 3.09666691868422826956235127849, 4.39613469036388071568375934230, 5.82990591357480902733840731133, 6.02546060014367813566146823396, 7.12694369805064815262071719534, 7.93556350227686936402798816626, 9.05838478867596469068078542910, 9.32490231627676220113297476012, 10.7495521486904629083810094781, 12.071063629605734143284750600916, 12.91760401111478026443561363555, 13.72042333385832896260767069975, 14.149973259572237889509960990160, 15.068493303916172778305346482243, 16.22694068822790261103086544350, 17.16185385397080490742044518671, 17.571699033202931778606285709408, 18.33755378030110238216811305852, 19.29308840114121805265460299551, 19.93891742957222994273147093646, 21.2696753726645993494444776620, 22.04185478240610651477711037312, 22.89878921471720124441364068862

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