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Properties

Label 1-51e2-2601.1885-r0-0-0
Degree $1$
Conductor $2601$
Sign $-0.988 - 0.148i$
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  + (−0.988 − 0.153i)2-s + (0.952 + 0.303i)4-s + (−0.198 − 0.980i)5-s + (0.0153 − 0.999i)7-s + (−0.895 − 0.445i)8-s + (0.0461 + 0.998i)10-s + (0.539 − 0.842i)11-s + (0.998 − 0.0615i)13-s + (−0.168 + 0.985i)14-s + (0.816 + 0.577i)16-s + (−0.361 + 0.932i)19-s + (0.107 − 0.994i)20-s + (−0.662 + 0.749i)22-s + (−0.858 − 0.513i)23-s + (−0.920 + 0.389i)25-s + (−0.995 − 0.0922i)26-s + ⋯
L(s)  = 1  + (−0.988 − 0.153i)2-s + (0.952 + 0.303i)4-s + (−0.198 − 0.980i)5-s + (0.0153 − 0.999i)7-s + (−0.895 − 0.445i)8-s + (0.0461 + 0.998i)10-s + (0.539 − 0.842i)11-s + (0.998 − 0.0615i)13-s + (−0.168 + 0.985i)14-s + (0.816 + 0.577i)16-s + (−0.361 + 0.932i)19-s + (0.107 − 0.994i)20-s + (−0.662 + 0.749i)22-s + (−0.858 − 0.513i)23-s + (−0.920 + 0.389i)25-s + (−0.995 − 0.0922i)26-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(2601\)    =    \(3^{2} \cdot 17^{2}\)
Sign: $-0.988 - 0.148i$
Analytic conductor: \(12.0789\)
Root analytic conductor: \(12.0789\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2601} (1885, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2601,\ (0:\ ),\ -0.988 - 0.148i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05682562882 - 0.7604863513i\)
\(L(\frac12)\) \(\approx\) \(0.05682562882 - 0.7604863513i\)
\(L(1)\) \(\approx\) \(0.5937782663 - 0.3327526663i\)
\(L(1)\) \(\approx\) \(0.5937782663 - 0.3327526663i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.988 - 0.153i)T \)
5 \( 1 + (-0.198 - 0.980i)T \)
7 \( 1 + (0.0153 - 0.999i)T \)
11 \( 1 + (0.539 - 0.842i)T \)
13 \( 1 + (0.998 - 0.0615i)T \)
19 \( 1 + (-0.361 + 0.932i)T \)
23 \( 1 + (-0.858 - 0.513i)T \)
29 \( 1 + (0.842 + 0.539i)T \)
31 \( 1 + (0.662 - 0.749i)T \)
37 \( 1 + (0.565 - 0.824i)T \)
41 \( 1 + (-0.926 + 0.375i)T \)
43 \( 1 + (-0.417 + 0.908i)T \)
47 \( 1 + (-0.969 - 0.243i)T \)
53 \( 1 + (-0.961 - 0.273i)T \)
59 \( 1 + (0.473 - 0.881i)T \)
61 \( 1 + (0.957 + 0.288i)T \)
67 \( 1 + (-0.153 - 0.988i)T \)
71 \( 1 + (-0.486 - 0.873i)T \)
73 \( 1 + (-0.638 - 0.769i)T \)
79 \( 1 + (0.994 + 0.107i)T \)
83 \( 1 + (0.920 - 0.389i)T \)
89 \( 1 + (-0.445 - 0.895i)T \)
97 \( 1 + (-0.999 - 0.0153i)T \)
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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

−19.36450274316050221842130154696, −19.04911790770471046689877808794, −18.09322716822661446370998973936, −17.87295947427373420105745300338, −17.12013193395433362928041235348, −15.914763116151302478438924451844, −15.60803389098570381985653599935, −14.98624605367485197032960048854, −14.28924084576718532537407262997, −13.33894112202103442771288216510, −12.10748040451761783739683532850, −11.71600829827000494684465063749, −11.05330860108654972858723756896, −10.13260571463972329248904962778, −9.67172549087405652784902552209, −8.65744733224058062585603489843, −8.25132098606188848735531688030, −7.21602840428967911129549263645, −6.52987712937979032581000229208, −6.1119659119607413216147474564, −5.00298574267620430815275996179, −3.80606390632561882746110178977, −2.851821898731092531372049292479, −2.1951836251532282790597997023, −1.30270227586255697980261086219, 0.3698189829055926452042496287, 1.17482492878375562403462239742, 1.84423544440001612897345675652, 3.31958665693605782763714527967, 3.83483699590400939627046329679, 4.7698438289982807659463365971, 6.15108715581779116368619444203, 6.3751182280248981844035198606, 7.6589686993492420540255660300, 8.24300005296270135558938936810, 8.65241213168529520452165738121, 9.64405527425289619126059646541, 10.239103515611735382877305821225, 11.09586625275779348426817998301, 11.63230423178183532932149921494, 12.47820552352061745388744492027, 13.21603851999265361046629016714, 13.98254002192862678024030104098, 14.86073843200472793234541120702, 16.01297469007365720334717915688, 16.348274860546105578616230148378, 16.78408263279505786704030705708, 17.61036478941958492270553712555, 18.26146092019766480651960833741, 19.18562987083010453615974570621

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