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Properties

Label 1-51e2-2601.178-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: $\chi_{2601} (178, \cdot )$
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

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

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