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Properties

Label 1-51e2-2601.218-r0-0-0
Degree $1$
Conductor $2601$
Sign $-0.269 - 0.963i$
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  + (−0.375 − 0.926i)2-s + (−0.717 + 0.696i)4-s + (−0.281 − 0.959i)5-s + (−0.733 − 0.679i)7-s + (0.914 + 0.403i)8-s + (−0.783 + 0.620i)10-s + (−0.802 + 0.596i)11-s + (−0.988 − 0.153i)13-s + (−0.354 + 0.935i)14-s + (0.0307 − 0.999i)16-s + (0.138 + 0.990i)19-s + (0.869 + 0.493i)20-s + (0.854 + 0.519i)22-s + (0.975 + 0.221i)23-s + (−0.842 + 0.539i)25-s + (0.228 + 0.973i)26-s + ⋯
L(s)  = 1  + (−0.375 − 0.926i)2-s + (−0.717 + 0.696i)4-s + (−0.281 − 0.959i)5-s + (−0.733 − 0.679i)7-s + (0.914 + 0.403i)8-s + (−0.783 + 0.620i)10-s + (−0.802 + 0.596i)11-s + (−0.988 − 0.153i)13-s + (−0.354 + 0.935i)14-s + (0.0307 − 0.999i)16-s + (0.138 + 0.990i)19-s + (0.869 + 0.493i)20-s + (0.854 + 0.519i)22-s + (0.975 + 0.221i)23-s + (−0.842 + 0.539i)25-s + (0.228 + 0.973i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.269 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.269 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2601\)    =    \(3^{2} \cdot 17^{2}\)
Sign: $-0.269 - 0.963i$
Analytic conductor: \(12.0789\)
Root analytic conductor: \(12.0789\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2601} (218, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2601,\ (0:\ ),\ -0.269 - 0.963i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4047696464 - 0.5335686993i\)
\(L(\frac12)\) \(\approx\) \(0.4047696464 - 0.5335686993i\)
\(L(1)\) \(\approx\) \(0.5315222204 - 0.3354776083i\)
\(L(1)\) \(\approx\) \(0.5315222204 - 0.3354776083i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.375 - 0.926i)T \)
5 \( 1 + (-0.281 - 0.959i)T \)
7 \( 1 + (-0.733 - 0.679i)T \)
11 \( 1 + (-0.802 + 0.596i)T \)
13 \( 1 + (-0.988 - 0.153i)T \)
19 \( 1 + (0.138 + 0.990i)T \)
23 \( 1 + (0.975 + 0.221i)T \)
29 \( 1 + (0.145 - 0.989i)T \)
31 \( 1 + (-0.519 + 0.854i)T \)
37 \( 1 + (-0.754 + 0.656i)T \)
41 \( 1 + (0.820 - 0.571i)T \)
43 \( 1 + (0.288 + 0.957i)T \)
47 \( 1 + (-0.577 - 0.816i)T \)
53 \( 1 + (0.638 + 0.769i)T \)
59 \( 1 + (0.901 - 0.431i)T \)
61 \( 1 + (0.744 - 0.667i)T \)
67 \( 1 + (0.389 + 0.920i)T \)
71 \( 1 + (0.466 - 0.884i)T \)
73 \( 1 + (-0.811 + 0.584i)T \)
79 \( 1 + (0.963 - 0.266i)T \)
83 \( 1 + (-0.539 - 0.842i)T \)
89 \( 1 + (-0.361 + 0.932i)T \)
97 \( 1 + (-0.0384 - 0.999i)T \)
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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

−19.30122575939512872021323514406, −18.830493656483483459164374109332, −18.14647024758744369480178833400, −17.55785684487331126349474250415, −16.57533545571290285983113498853, −15.99881727872235977837378446758, −15.350111795171193991994647070414, −14.803555533310873864193001902316, −14.140729590198254461610151562008, −13.21617371767449229088701226947, −12.65685970748983883743873820570, −11.49943143909913701327915396334, −10.781219804974452713761640459183, −10.076147065399094873596382573296, −9.29156225586371934612306485535, −8.68187256999807774727620943765, −7.70463206639428246530096335478, −7.06624648227858647861215614468, −6.54786793934466701340511167593, −5.57543830533610579969858835790, −5.06411672682989450830321983137, −3.88174509188245995241025532124, −2.90533513432948965195064319656, −2.269517940540566685986438091834, −0.560162118346319590963971885827, 0.46155588010477035312050928158, 1.428606568664897333489127040571, 2.41004860063260345527389602156, 3.312531595478332114967498735039, 4.095998553222236233647558282722, 4.85172763913594793579079002978, 5.511521749446899520502391828888, 6.983917094091301910702233941033, 7.62103849158290737485264983683, 8.280127639658159622568656256390, 9.19656681103764750359389553582, 9.88795246996436726771874403530, 10.27727137285724334520145196124, 11.24523065519003084317387464519, 12.15223512500833094719842997418, 12.62614840882781961166363987192, 13.13334402117693737392058740702, 13.8732523777259303065818510630, 14.870603482605787980768021798682, 15.8943214256593890979712942238, 16.45855673963823438240796525626, 17.22203572669598064206645857102, 17.60589683464030121774029239489, 18.68898897695568095557849036444, 19.35121889455002159763014336108

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