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Properties

Label 1-5e3-125.64-r0-0-0
Degree $1$
Conductor $125$
Sign $0.0376 + 0.999i$
Analytic cond. $0.580497$
Root an. cond. $0.580497$
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  + (0.929 + 0.368i)2-s + (−0.876 + 0.481i)3-s + (0.728 + 0.684i)4-s + (−0.992 + 0.125i)6-s + (0.809 + 0.587i)7-s + (0.425 + 0.904i)8-s + (0.535 − 0.844i)9-s + (−0.929 − 0.368i)11-s + (−0.968 − 0.248i)12-s + (−0.535 + 0.844i)13-s + (0.535 + 0.844i)14-s + (0.0627 + 0.998i)16-s + (−0.728 + 0.684i)17-s + (0.809 − 0.587i)18-s + (0.876 + 0.481i)19-s + ⋯
L(s)  = 1  + (0.929 + 0.368i)2-s + (−0.876 + 0.481i)3-s + (0.728 + 0.684i)4-s + (−0.992 + 0.125i)6-s + (0.809 + 0.587i)7-s + (0.425 + 0.904i)8-s + (0.535 − 0.844i)9-s + (−0.929 − 0.368i)11-s + (−0.968 − 0.248i)12-s + (−0.535 + 0.844i)13-s + (0.535 + 0.844i)14-s + (0.0627 + 0.998i)16-s + (−0.728 + 0.684i)17-s + (0.809 − 0.587i)18-s + (0.876 + 0.481i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0376 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0376 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(125\)    =    \(5^{3}\)
Sign: $0.0376 + 0.999i$
Analytic conductor: \(0.580497\)
Root analytic conductor: \(0.580497\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{125} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 125,\ (0:\ ),\ 0.0376 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.025484560 + 0.9875355315i\)
\(L(\frac12)\) \(\approx\) \(1.025484560 + 0.9875355315i\)
\(L(1)\) \(\approx\) \(1.215445941 + 0.6557256011i\)
\(L(1)\) \(\approx\) \(1.215445941 + 0.6557256011i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 + (0.929 + 0.368i)T \)
3 \( 1 + (-0.876 + 0.481i)T \)
7 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 + (-0.929 - 0.368i)T \)
13 \( 1 + (-0.535 + 0.844i)T \)
17 \( 1 + (-0.728 + 0.684i)T \)
19 \( 1 + (0.876 + 0.481i)T \)
23 \( 1 + (0.637 - 0.770i)T \)
29 \( 1 + (-0.187 - 0.982i)T \)
31 \( 1 + (0.728 - 0.684i)T \)
37 \( 1 + (-0.0627 - 0.998i)T \)
41 \( 1 + (-0.637 - 0.770i)T \)
43 \( 1 + (-0.309 - 0.951i)T \)
47 \( 1 + (0.425 - 0.904i)T \)
53 \( 1 + (0.992 + 0.125i)T \)
59 \( 1 + (0.968 + 0.248i)T \)
61 \( 1 + (-0.637 + 0.770i)T \)
67 \( 1 + (0.187 - 0.982i)T \)
71 \( 1 + (-0.425 + 0.904i)T \)
73 \( 1 + (-0.968 + 0.248i)T \)
79 \( 1 + (0.876 - 0.481i)T \)
83 \( 1 + (-0.876 - 0.481i)T \)
89 \( 1 + (0.968 - 0.248i)T \)
97 \( 1 + (0.187 + 0.982i)T \)
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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

−28.98962057155059403548631328968, −27.95120358283948216392468054153, −26.91894682799676396145809779765, −25.16641472604483015770972979344, −24.278454784959195557953129909525, −23.527034395484320257523506610993, −22.70577269271266433696485352482, −21.765264505312017613520758124907, −20.62005488693106892833173306432, −19.74692687225436774951070713233, −18.28975113769780780520260054553, −17.50023792946863049593108223086, −16.11008500817027070524950848907, −15.104834626038505937235958901635, −13.69662150087929364713293723518, −12.99365365250670467420191349265, −11.8044160524607267184679964838, −10.97917423164582937219639690341, −10.0487784236661250077127097266, −7.6947065710414001742682405145, −6.8714338458701787199562852251, −5.22384808611799825941660372242, −4.8095874868558603281488236857, −2.8155052750332962697369480996, −1.2471604189759904933118423347, 2.27512042144352539047188868181, 4.065107309413524928491268703487, 5.08999034635453267834084840244, 5.9305308118263982728585918095, 7.26335372523374004166222835953, 8.66805591349231292539565174290, 10.435728890000091261247354161590, 11.491514399851142324548493287481, 12.21415863582210299251224453024, 13.515490850364795497846096220779, 14.83530013909913938162972179830, 15.57785316651553409643970607250, 16.62742280054242959451294065780, 17.54050106147067681210857909373, 18.74286372863868605918609057473, 20.603274263840307122466853142604, 21.32374002175541569017517623771, 22.066998941248936116533595444585, 23.06800510768908370100386275177, 24.1226665985023725373743536925, 24.600615823036640277137417677102, 26.29580900670062935101892075620, 26.92648810053618158288204550578, 28.520526506876013479113076439363, 28.96317322147509274080174361131

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