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Properties

Label 1-6025-6025.378-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.997 - 0.0728i$
Analytic cond. $27.9799$
Root an. cond. $27.9799$
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  + (0.358 − 0.933i)2-s + (−0.544 + 0.838i)3-s + (−0.743 − 0.669i)4-s + (0.587 + 0.809i)6-s + (0.430 − 0.902i)7-s + (−0.891 + 0.453i)8-s + (−0.406 − 0.913i)9-s + (−0.688 + 0.725i)11-s + (0.965 − 0.258i)12-s + (0.991 − 0.130i)13-s + (−0.688 − 0.725i)14-s + (0.104 + 0.994i)16-s + (−0.852 + 0.522i)17-s + (−0.998 + 0.0523i)18-s + (−0.430 + 0.902i)19-s + ⋯
L(s)  = 1  + (0.358 − 0.933i)2-s + (−0.544 + 0.838i)3-s + (−0.743 − 0.669i)4-s + (0.587 + 0.809i)6-s + (0.430 − 0.902i)7-s + (−0.891 + 0.453i)8-s + (−0.406 − 0.913i)9-s + (−0.688 + 0.725i)11-s + (0.965 − 0.258i)12-s + (0.991 − 0.130i)13-s + (−0.688 − 0.725i)14-s + (0.104 + 0.994i)16-s + (−0.852 + 0.522i)17-s + (−0.998 + 0.0523i)18-s + (−0.430 + 0.902i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.997 - 0.0728i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (378, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.997 - 0.0728i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.115180701 - 0.04068327042i\)
\(L(\frac12)\) \(\approx\) \(1.115180701 - 0.04068327042i\)
\(L(1)\) \(\approx\) \(0.8572981616 - 0.2590740372i\)
\(L(1)\) \(\approx\) \(0.8572981616 - 0.2590740372i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.358 - 0.933i)T \)
3 \( 1 + (-0.544 + 0.838i)T \)
7 \( 1 + (0.430 - 0.902i)T \)
11 \( 1 + (-0.688 + 0.725i)T \)
13 \( 1 + (0.991 - 0.130i)T \)
17 \( 1 + (-0.852 + 0.522i)T \)
19 \( 1 + (-0.430 + 0.902i)T \)
23 \( 1 + (0.760 - 0.649i)T \)
29 \( 1 + (-0.777 - 0.629i)T \)
31 \( 1 + (-0.999 + 0.0261i)T \)
37 \( 1 + (-0.430 + 0.902i)T \)
41 \( 1 + (0.453 - 0.891i)T \)
43 \( 1 + (0.0784 - 0.996i)T \)
47 \( 1 + (0.987 + 0.156i)T \)
53 \( 1 + (0.933 - 0.358i)T \)
59 \( 1 + (-0.0523 + 0.998i)T \)
61 \( 1 + (-0.891 - 0.453i)T \)
67 \( 1 + (-0.358 + 0.933i)T \)
71 \( 1 + (0.284 - 0.958i)T \)
73 \( 1 + (-0.382 - 0.923i)T \)
79 \( 1 + (0.156 + 0.987i)T \)
83 \( 1 + (-0.978 - 0.207i)T \)
89 \( 1 + (0.999 - 0.0261i)T \)
97 \( 1 + (0.913 + 0.406i)T \)
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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

−17.79367031302136694175192141696, −17.07053583558389251240356275898, −16.30387414195767968861961080881, −15.81846892056962661906291582483, −15.21869808083963035974316108151, −14.40962358233825157484578817981, −13.69886042797083967763440163365, −13.07162362443644603336260891093, −12.83065793830224139564763668803, −11.845339400925913651700325618465, −11.10936928668984003039859399624, −10.908099326304851966010964171951, −9.2664158835449052894656762212, −8.86609711608141073020785102437, −8.26513039818040172789742608751, −7.47005419737582797459109802820, −6.94682380333416810348428744715, −6.105763765964353054766572015988, −5.6285691569415723167404335665, −5.103728282074447632648385791529, −4.31510521685478298579461139847, −3.17652975170718512285438611566, −2.533705212323734787211589824516, −1.53856914649461832301891962944, −0.39609638864129361860455534371, 0.67154496168213731264920516593, 1.62744338225341871647637681155, 2.40858471565754706570648714854, 3.55576742401965753610711284689, 3.97074037506634895615256496823, 4.53793647726985050502555115157, 5.280237389450442187286191346883, 5.921018506013811156880904652538, 6.72595893222528579921547415206, 7.72152708346769345334485644533, 8.693867209696277882910591141082, 9.12577654549684141493088504862, 10.20253939247053031139381088468, 10.54961742744465766568823691862, 10.85294620270621252917085673324, 11.640033881356318247059055322911, 12.37603182978787563610363003342, 13.064511330746442001286640770406, 13.60747932388085186323291050211, 14.45935676036638114780137426489, 15.09197146740912883692947876525, 15.50698673347858297365429690576, 16.50325170399731185179400273818, 17.14871139980961550024091969875, 17.68384149492627527462213815029

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