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Properties

Label 1-925-925.59-r1-0-0
Degree $1$
Conductor $925$
Sign $-0.418 + 0.908i$
Analytic cond. $99.4050$
Root an. cond. $99.4050$
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  + (−0.927 − 0.374i)2-s + (−0.241 + 0.970i)3-s + (0.719 + 0.694i)4-s + (0.587 − 0.809i)6-s + (0.939 + 0.342i)7-s + (−0.406 − 0.913i)8-s + (−0.882 − 0.469i)9-s + (0.978 + 0.207i)11-s + (−0.848 + 0.529i)12-s + (0.139 − 0.990i)13-s + (−0.743 − 0.669i)14-s + (0.0348 + 0.999i)16-s + (0.694 + 0.719i)17-s + (0.642 + 0.766i)18-s + (−0.0697 − 0.997i)19-s + ⋯
L(s)  = 1  + (−0.927 − 0.374i)2-s + (−0.241 + 0.970i)3-s + (0.719 + 0.694i)4-s + (0.587 − 0.809i)6-s + (0.939 + 0.342i)7-s + (−0.406 − 0.913i)8-s + (−0.882 − 0.469i)9-s + (0.978 + 0.207i)11-s + (−0.848 + 0.529i)12-s + (0.139 − 0.990i)13-s + (−0.743 − 0.669i)14-s + (0.0348 + 0.999i)16-s + (0.694 + 0.719i)17-s + (0.642 + 0.766i)18-s + (−0.0697 − 0.997i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.418 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.418 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $-0.418 + 0.908i$
Analytic conductor: \(99.4050\)
Root analytic conductor: \(99.4050\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 925,\ (1:\ ),\ -0.418 + 0.908i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5358570621 + 0.8371067085i\)
\(L(\frac12)\) \(\approx\) \(0.5358570621 + 0.8371067085i\)
\(L(1)\) \(\approx\) \(0.6977022777 + 0.1745836498i\)
\(L(1)\) \(\approx\) \(0.6977022777 + 0.1745836498i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.927 - 0.374i)T \)
3 \( 1 + (-0.241 + 0.970i)T \)
7 \( 1 + (0.939 + 0.342i)T \)
11 \( 1 + (0.978 + 0.207i)T \)
13 \( 1 + (0.139 - 0.990i)T \)
17 \( 1 + (0.694 + 0.719i)T \)
19 \( 1 + (-0.0697 - 0.997i)T \)
23 \( 1 + (-0.743 - 0.669i)T \)
29 \( 1 + (-0.994 + 0.104i)T \)
31 \( 1 + (-0.587 + 0.809i)T \)
41 \( 1 + (-0.990 - 0.139i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.913 - 0.406i)T \)
53 \( 1 + (-0.559 + 0.829i)T \)
59 \( 1 + (-0.999 + 0.0348i)T \)
61 \( 1 + (0.469 + 0.882i)T \)
67 \( 1 + (0.559 + 0.829i)T \)
71 \( 1 + (-0.241 + 0.970i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (0.275 + 0.961i)T \)
83 \( 1 + (0.719 - 0.694i)T \)
89 \( 1 + (-0.788 + 0.615i)T \)
97 \( 1 + (0.406 - 0.913i)T \)
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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

−21.1649709064842456175267577922, −20.3928944054339856759698593663, −19.62436589932990320729742990573, −18.80100892062234059457756583872, −18.32853542006788229279395808549, −17.5016166493931181266320759240, −16.72780129002141581973772534349, −16.3946437424932938812139247532, −14.87367443138706356356472375683, −14.27626346873326793276352095878, −13.69767663986593503005319659759, −12.2435013168959398005015614692, −11.495842555426252599268628855803, −11.168238265191537322419448632853, −9.83256104290033052412212222279, −9.05808089780044091999284269689, −7.97561271956484536596554037654, −7.62631297341765208172426349760, −6.61078561480142267642444225803, −5.94246148536690275962578878507, −4.92840499504182083296293089112, −3.49505858431665177253039668718, −1.75763783384445460319459557010, −1.64115491233879281378945082507, −0.34342866309464827779094207968, 0.96304441224417230497028102388, 2.07431408390632773086920031989, 3.26888786467443438464960060631, 4.06759106607820470479737004057, 5.20367305418808792271378221846, 6.12312290422530435337604482659, 7.26889245866864990647357079916, 8.39373111730218515980595208298, 8.80450269313965253697986932888, 9.80969892154336584062892323597, 10.53041168253898048906228450129, 11.22464160695276321796932964241, 11.93704100133969833320757667472, 12.71929062290899224733445881283, 14.227468423970385751418305646150, 15.02477129576709561703634216629, 15.57707948570980658015886009199, 16.63628350835505374453671107083, 17.2267081060802098059525423746, 17.829727961467716377063862137860, 18.64431796998376688093169574334, 19.82469545445068134342740990764, 20.27396956455196269370417953259, 21.01133440253363342635515979935, 21.91161380890986044057170433872

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