Properties

Label 1-51e2-2601.4-r0-0-0
Degree $1$
Conductor $2601$
Sign $0.148 - 0.988i$
Analytic cond. $12.0789$
Root an. cond. $12.0789$
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  + (0.153 − 0.988i)2-s + (−0.952 − 0.303i)4-s + (−0.833 − 0.552i)5-s + (0.717 − 0.696i)7-s + (−0.445 + 0.895i)8-s + (−0.673 + 0.739i)10-s + (−0.976 + 0.213i)11-s + (−0.998 + 0.0615i)13-s + (−0.577 − 0.816i)14-s + (0.816 + 0.577i)16-s + (−0.932 − 0.361i)19-s + (0.626 + 0.779i)20-s + (0.0615 + 0.998i)22-s + (0.243 + 0.969i)23-s + (0.389 + 0.920i)25-s + (−0.0922 + 0.995i)26-s + ⋯
L(s)  = 1  + (0.153 − 0.988i)2-s + (−0.952 − 0.303i)4-s + (−0.833 − 0.552i)5-s + (0.717 − 0.696i)7-s + (−0.445 + 0.895i)8-s + (−0.673 + 0.739i)10-s + (−0.976 + 0.213i)11-s + (−0.998 + 0.0615i)13-s + (−0.577 − 0.816i)14-s + (0.816 + 0.577i)16-s + (−0.932 − 0.361i)19-s + (0.626 + 0.779i)20-s + (0.0615 + 0.998i)22-s + (0.243 + 0.969i)23-s + (0.389 + 0.920i)25-s + (−0.0922 + 0.995i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.148 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.148 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2601\)    =    \(3^{2} \cdot 17^{2}\)
Sign: $0.148 - 0.988i$
Analytic conductor: \(12.0789\)
Root analytic conductor: \(12.0789\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2601} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2601,\ (0:\ ),\ 0.148 - 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7286926060 - 0.6273643090i\)
\(L(\frac12)\) \(\approx\) \(0.7286926060 - 0.6273643090i\)
\(L(1)\) \(\approx\) \(0.6734783505 - 0.4622386370i\)
\(L(1)\) \(\approx\) \(0.6734783505 - 0.4622386370i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.153 - 0.988i)T \)
5 \( 1 + (-0.833 - 0.552i)T \)
7 \( 1 + (0.717 - 0.696i)T \)
11 \( 1 + (-0.976 + 0.213i)T \)
13 \( 1 + (-0.998 + 0.0615i)T \)
19 \( 1 + (-0.932 - 0.361i)T \)
23 \( 1 + (0.243 + 0.969i)T \)
29 \( 1 + (0.976 - 0.213i)T \)
31 \( 1 + (0.0615 + 0.998i)T \)
37 \( 1 + (0.183 + 0.982i)T \)
41 \( 1 + (-0.920 - 0.389i)T \)
43 \( 1 + (0.908 + 0.417i)T \)
47 \( 1 + (0.969 + 0.243i)T \)
53 \( 1 + (0.273 - 0.961i)T \)
59 \( 1 + (-0.881 - 0.473i)T \)
61 \( 1 + (0.473 + 0.881i)T \)
67 \( 1 + (-0.153 - 0.988i)T \)
71 \( 1 + (0.961 + 0.273i)T \)
73 \( 1 + (-0.995 - 0.0922i)T \)
79 \( 1 + (-0.626 - 0.779i)T \)
83 \( 1 + (0.389 + 0.920i)T \)
89 \( 1 + (0.445 + 0.895i)T \)
97 \( 1 + (-0.717 + 0.696i)T \)
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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

−19.17100744548319260210697057522, −18.66637048180656923122141841971, −18.191726954913907133421111169623, −17.29230756630004801604792725689, −16.68513636662948146737841701445, −15.61933067068045470991017653552, −15.46495370775904180294916609427, −14.505345754099935285537086808720, −14.34125160788227028617744419125, −13.10166374858992628134931645587, −12.407485310515575634268784153726, −11.85537589573323121301731340986, −10.76507321923076309731597352548, −10.21847560959534613105968515426, −9.02775875314150831761491572634, −8.375572684712560226874739099084, −7.78642261426780630206196953802, −7.20158191640490856716018494687, −6.268498959029168593055255329403, −5.5130275101954243618361251607, −4.66840672635075339506795436721, −4.15275294620787805861461892812, −2.921875996152832383468630993972, −2.31224629195793081159899279745, −0.50458575397582341389862826890, 0.61599485989979455135452401012, 1.572785776707172313770836647708, 2.52639538616998360053399447649, 3.39988591180719478734387489833, 4.376512965382448200910873608187, 4.787368610396038457170865896677, 5.40778164806562648002032043911, 6.87957267253616642386951933535, 7.74544446504425003679417142108, 8.279640994408592644692401080406, 9.07632466106577426533242184159, 10.052227343395935939581832923402, 10.62337426236602236149531370531, 11.326421598385385240100298540258, 12.04354654403721764591388708009, 12.61694636262090354456624067230, 13.37117448450120865464200809616, 14.02614851390669482287150864752, 14.96050267344357599356835553409, 15.43397916297767472057056603916, 16.49718605643600412862686036462, 17.401696816562772803585328442710, 17.666347993956732887821977543025, 18.78081082953154835280181318837, 19.38782061886551049171359728301

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