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Properties

Label 1-51e2-2601.1786-r0-0-0
Degree $1$
Conductor $2601$
Sign $-0.786 - 0.617i$
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  + (−0.389 − 0.920i)2-s + (−0.696 + 0.717i)4-s + (−0.779 − 0.626i)5-s + (−0.908 − 0.417i)7-s + (0.932 + 0.361i)8-s + (−0.273 + 0.961i)10-s + (0.969 + 0.243i)11-s + (−0.153 − 0.988i)13-s + (−0.0307 + 0.999i)14-s + (−0.0307 − 0.999i)16-s + (−0.602 − 0.798i)19-s + (0.992 − 0.122i)20-s + (−0.153 − 0.988i)22-s + (0.816 − 0.577i)23-s + (0.213 + 0.976i)25-s + (−0.850 + 0.526i)26-s + ⋯
L(s)  = 1  + (−0.389 − 0.920i)2-s + (−0.696 + 0.717i)4-s + (−0.779 − 0.626i)5-s + (−0.908 − 0.417i)7-s + (0.932 + 0.361i)8-s + (−0.273 + 0.961i)10-s + (0.969 + 0.243i)11-s + (−0.153 − 0.988i)13-s + (−0.0307 + 0.999i)14-s + (−0.0307 − 0.999i)16-s + (−0.602 − 0.798i)19-s + (0.992 − 0.122i)20-s + (−0.153 − 0.988i)22-s + (0.816 − 0.577i)23-s + (0.213 + 0.976i)25-s + (−0.850 + 0.526i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.786 - 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.786 - 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2601\)    =    \(3^{2} \cdot 17^{2}\)
Sign: $-0.786 - 0.617i$
Analytic conductor: \(12.0789\)
Root analytic conductor: \(12.0789\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2601} (1786, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2601,\ (0:\ ),\ -0.786 - 0.617i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3043369988 - 0.8805751534i\)
\(L(\frac12)\) \(\approx\) \(0.3043369988 - 0.8805751534i\)
\(L(1)\) \(\approx\) \(0.5717126204 - 0.4372253719i\)
\(L(1)\) \(\approx\) \(0.5717126204 - 0.4372253719i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.389 - 0.920i)T \)
5 \( 1 + (-0.779 - 0.626i)T \)
7 \( 1 + (-0.908 - 0.417i)T \)
11 \( 1 + (0.969 + 0.243i)T \)
13 \( 1 + (-0.153 - 0.988i)T \)
19 \( 1 + (-0.602 - 0.798i)T \)
23 \( 1 + (0.816 - 0.577i)T \)
29 \( 1 + (0.969 + 0.243i)T \)
31 \( 1 + (-0.153 - 0.988i)T \)
37 \( 1 + (0.445 + 0.895i)T \)
41 \( 1 + (0.213 - 0.976i)T \)
43 \( 1 + (0.881 - 0.473i)T \)
47 \( 1 + (0.816 + 0.577i)T \)
53 \( 1 + (0.0922 + 0.995i)T \)
59 \( 1 + (0.332 + 0.943i)T \)
61 \( 1 + (0.332 - 0.943i)T \)
67 \( 1 + (-0.389 + 0.920i)T \)
71 \( 1 + (0.0922 + 0.995i)T \)
73 \( 1 + (-0.850 + 0.526i)T \)
79 \( 1 + (0.992 - 0.122i)T \)
83 \( 1 + (0.213 + 0.976i)T \)
89 \( 1 + (0.932 - 0.361i)T \)
97 \( 1 + (-0.908 - 0.417i)T \)
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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

−19.29076828275504586470246478862, −19.13494343782377761859886107102, −18.24433037447743024078049741882, −17.49328490778202403061879296218, −16.48273847867356215358593957563, −16.29847927524733943073359682404, −15.45746485943311829315647068463, −14.69641982836645527921965433946, −14.3134011273247367913030346653, −13.41801419624955555985385198859, −12.45455087856096640618729124542, −11.77248226710731077602955796213, −10.8940404307174802607975736647, −10.11340294952170389922972531685, −9.247377706884174590900492834211, −8.81332953671358783750547877486, −7.88009707951393913380077440503, −7.0610728015614424246809485998, −6.493756143033544309411615323872, −6.011528647360451166764791839074, −4.80631960981567897866054149911, −3.978195382260774410047115448543, −3.32309587113179106463418668387, −2.08559197618823729736688043716, −0.845022715465618109691319877054, 0.557276142477183557978889995710, 1.0726565759306693335747782619, 2.507330994179917991828324105483, 3.17278495568341133680280259264, 4.14216954830001371488496050255, 4.46330967477498874883377714525, 5.63641176413091046472250479015, 6.84563209819525793059186980505, 7.4446796666940829164433771049, 8.40340269738963791038455007890, 8.992925602824310583538613348499, 9.63742895181607727620824620552, 10.51467074151218618478048263924, 11.09422909387432657031828484369, 12.01805612821392878514327406216, 12.54894406910175610986404500026, 13.06092402071295171816861911905, 13.82205464546873625847733442920, 14.89213776232327036905625933658, 15.64476793276237494236578120119, 16.454453209377308781829713379298, 17.18475787340632446931033694591, 17.466460565998204279131120976827, 18.742822981207527899879223236116, 19.22948859079916310299133459171

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