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Properties

Label 1-1625-1625.1162-r0-0-0
Degree $1$
Conductor $1625$
Sign $0.759 + 0.650i$
Analytic cond. $7.54646$
Root an. cond. $7.54646$
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  + (0.968 + 0.248i)2-s + (0.982 + 0.187i)3-s + (0.876 + 0.481i)4-s + (0.904 + 0.425i)6-s + (0.809 − 0.587i)7-s + (0.728 + 0.684i)8-s + (0.929 + 0.368i)9-s + (−0.248 + 0.968i)11-s + (0.770 + 0.637i)12-s + (0.929 − 0.368i)14-s + (0.535 + 0.844i)16-s + (−0.481 − 0.876i)17-s + (0.809 + 0.587i)18-s + (0.982 − 0.187i)19-s + (0.904 − 0.425i)21-s + (−0.481 + 0.876i)22-s + ⋯
L(s)  = 1  + (0.968 + 0.248i)2-s + (0.982 + 0.187i)3-s + (0.876 + 0.481i)4-s + (0.904 + 0.425i)6-s + (0.809 − 0.587i)7-s + (0.728 + 0.684i)8-s + (0.929 + 0.368i)9-s + (−0.248 + 0.968i)11-s + (0.770 + 0.637i)12-s + (0.929 − 0.368i)14-s + (0.535 + 0.844i)16-s + (−0.481 − 0.876i)17-s + (0.809 + 0.587i)18-s + (0.982 − 0.187i)19-s + (0.904 − 0.425i)21-s + (−0.481 + 0.876i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1625 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.759 + 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1625 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.759 + 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1625\)    =    \(5^{3} \cdot 13\)
Sign: $0.759 + 0.650i$
Analytic conductor: \(7.54646\)
Root analytic conductor: \(7.54646\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1625} (1162, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1625,\ (0:\ ),\ 0.759 + 0.650i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.660164455 + 1.724079902i\)
\(L(\frac12)\) \(\approx\) \(4.660164455 + 1.724079902i\)
\(L(1)\) \(\approx\) \(2.778312932 + 0.6734092058i\)
\(L(1)\) \(\approx\) \(2.778312932 + 0.6734092058i\)

Euler product

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

−20.45144615431826248118986543959, −19.7939370500055700807267842572, −19.03627375001822719515456232024, −18.39903662714436171697302331528, −17.51463575162262757079337277610, −16.13269461321741510777346659286, −15.7302006332911221316392819037, −14.90470794714750592526217507361, −14.17981644491929567682499614809, −13.810853548455378926329457734989, −12.92169791541277116059949498282, −12.168945133585671892384031953997, −11.47561740383533940730324884658, −10.56118846622422963258113601919, −9.80563960722262536633254217994, −8.55494747458209560416428370982, −8.21032183010183799680374610608, −7.15818348718933678802295000131, −6.261429188725244391335491498984, −5.39782851810871937556675538701, −4.562935624649398241456446934065, −3.58134353857051213428196058957, −2.96022143990940860163335517006, −1.98481149849907954621408275822, −1.32234241780921788089591099078, 1.48226467823618809878832076210, 2.28665207473366309260452285678, 3.10304545258559695873510445332, 4.181150104256755470249318674574, 4.58383218837937373208706665866, 5.44112927041141880121536112259, 6.73881288822179909514631593162, 7.539027690361668052254162463226, 7.81523136113617479235947472518, 8.95267265745830600656339200130, 9.91965430622273853602281576839, 10.675956338908150127995676309399, 11.63093201666758118951925328877, 12.34510743828923785986637444708, 13.361294540663779484619349876731, 13.81037911352864603205717378605, 14.45933299758323140084712911013, 15.11740064901767178995607931028, 15.86198403737497907574784897780, 16.41685591544899602536899901018, 17.69206383479543978500748236908, 18.046547812258570601946476474085, 19.446988973644424738280042134334, 20.05491793793281692363273236541, 20.74800258990701113773470862690

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