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Properties

Label 1-5e4-625.121-r0-0-0
Degree $1$
Conductor $625$
Sign $-0.434 + 0.900i$
Analytic cond. $2.90248$
Root an. cond. $2.90248$
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  + (0.823 − 0.567i)2-s + (−0.137 − 0.990i)3-s + (0.356 − 0.934i)4-s + (−0.675 − 0.737i)6-s + (−0.929 + 0.368i)7-s + (−0.236 − 0.971i)8-s + (−0.962 + 0.272i)9-s + (−0.332 + 0.942i)11-s + (−0.974 − 0.224i)12-s + (−0.556 − 0.830i)13-s + (−0.556 + 0.830i)14-s + (−0.745 − 0.666i)16-s + (−0.778 + 0.627i)17-s + (−0.637 + 0.770i)18-s + (0.693 − 0.720i)19-s + ⋯
L(s)  = 1  + (0.823 − 0.567i)2-s + (−0.137 − 0.990i)3-s + (0.356 − 0.934i)4-s + (−0.675 − 0.737i)6-s + (−0.929 + 0.368i)7-s + (−0.236 − 0.971i)8-s + (−0.962 + 0.272i)9-s + (−0.332 + 0.942i)11-s + (−0.974 − 0.224i)12-s + (−0.556 − 0.830i)13-s + (−0.556 + 0.830i)14-s + (−0.745 − 0.666i)16-s + (−0.778 + 0.627i)17-s + (−0.637 + 0.770i)18-s + (0.693 − 0.720i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $-0.434 + 0.900i$
Analytic conductor: \(2.90248\)
Root analytic conductor: \(2.90248\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{625} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 625,\ (0:\ ),\ -0.434 + 0.900i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2977407228 - 0.4744186264i\)
\(L(\frac12)\) \(\approx\) \(-0.2977407228 - 0.4744186264i\)
\(L(1)\) \(\approx\) \(0.7326264831 - 0.6917034638i\)
\(L(1)\) \(\approx\) \(0.7326264831 - 0.6917034638i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 + (0.823 - 0.567i)T \)
3 \( 1 + (-0.137 - 0.990i)T \)
7 \( 1 + (-0.929 + 0.368i)T \)
11 \( 1 + (-0.332 + 0.942i)T \)
13 \( 1 + (-0.556 - 0.830i)T \)
17 \( 1 + (-0.778 + 0.627i)T \)
19 \( 1 + (0.693 - 0.720i)T \)
23 \( 1 + (-0.711 - 0.702i)T \)
29 \( 1 + (-0.947 + 0.320i)T \)
31 \( 1 + (-0.778 + 0.627i)T \)
37 \( 1 + (-0.863 + 0.503i)T \)
41 \( 1 + (0.162 - 0.986i)T \)
43 \( 1 + (-0.187 + 0.982i)T \)
47 \( 1 + (-0.379 - 0.925i)T \)
53 \( 1 + (0.112 - 0.993i)T \)
59 \( 1 + (0.656 + 0.754i)T \)
61 \( 1 + (0.162 + 0.986i)T \)
67 \( 1 + (0.0125 - 0.999i)T \)
71 \( 1 + (0.762 - 0.647i)T \)
73 \( 1 + (0.920 + 0.391i)T \)
79 \( 1 + (-0.137 - 0.990i)T \)
83 \( 1 + (-0.470 - 0.882i)T \)
89 \( 1 + (-0.974 + 0.224i)T \)
97 \( 1 + (-0.597 - 0.801i)T \)
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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

−23.46202207313503387113214425852, −22.45439223496454566310609310787, −22.15899201231287433528846357755, −21.30974364189149591027556112356, −20.47475331545315185888541599695, −19.73695500889282182648511243012, −18.53208213368647870908882273533, −17.26422964703373763838241793791, −16.56566978620873080618933352936, −16.02571543867841792982709614822, −15.40912393225654924093287108136, −14.20781350036293780229271144199, −13.79479382743881237735646533410, −12.74083488836704797583252690734, −11.65101366195607042093738362956, −11.05042259363810528134467670239, −9.78161304772567482242827137001, −9.108179983122424731120348825640, −7.91466980902507151361235838255, −6.88967036280608309049524796629, −5.915314977349393121213804816608, −5.229624103091356672551252173513, −4.03712323882946353750583966163, −3.51526845278901266773060236323, −2.43350731300214502149553135173, 0.19145754283296115031489705058, 1.82206335694611695803719736525, 2.56719764874900608329120105189, 3.52383167838868675258956881446, 4.949701096509330185380054502013, 5.704206666941835030306791428706, 6.70813753166630686537968428804, 7.318614469492030213560032954601, 8.71996189000759062341171793300, 9.825102047954127087394642023880, 10.65121962085239815283875636137, 11.718689034338788838051576960740, 12.592211797226840016144012953845, 12.86225672782185336041862219385, 13.7190672794436606709767268996, 14.8155154924084985036682127588, 15.46512008183319647541735350604, 16.52425277663372421719328502636, 17.825600984080407460994405058, 18.306796015687474983925071478198, 19.46191205980672408222816362384, 19.84705322347147200132284950858, 20.57592619129041078004171894262, 21.96631177477043170048891886716, 22.48033409629733534178638248190

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