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Properties

Label 1-469-469.5-r0-0-0
Degree $1$
Conductor $469$
Sign $-0.633 + 0.773i$
Analytic cond. $2.17802$
Root an. cond. $2.17802$
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  + (0.786 − 0.618i)2-s + (−0.327 − 0.945i)3-s + (0.235 − 0.971i)4-s + (−0.888 − 0.458i)5-s + (−0.841 − 0.540i)6-s + (−0.415 − 0.909i)8-s + (−0.786 + 0.618i)9-s + (−0.981 + 0.189i)10-s + (−0.0475 − 0.998i)11-s + (−0.995 + 0.0950i)12-s + (0.415 − 0.909i)13-s + (−0.142 + 0.989i)15-s + (−0.888 − 0.458i)16-s + (−0.723 + 0.690i)17-s + (−0.235 + 0.971i)18-s + (−0.928 + 0.371i)19-s + ⋯
L(s)  = 1  + (0.786 − 0.618i)2-s + (−0.327 − 0.945i)3-s + (0.235 − 0.971i)4-s + (−0.888 − 0.458i)5-s + (−0.841 − 0.540i)6-s + (−0.415 − 0.909i)8-s + (−0.786 + 0.618i)9-s + (−0.981 + 0.189i)10-s + (−0.0475 − 0.998i)11-s + (−0.995 + 0.0950i)12-s + (0.415 − 0.909i)13-s + (−0.142 + 0.989i)15-s + (−0.888 − 0.458i)16-s + (−0.723 + 0.690i)17-s + (−0.235 + 0.971i)18-s + (−0.928 + 0.371i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.633 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.633 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(469\)    =    \(7 \cdot 67\)
Sign: $-0.633 + 0.773i$
Analytic conductor: \(2.17802\)
Root analytic conductor: \(2.17802\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{469} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 469,\ (0:\ ),\ -0.633 + 0.773i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4332834622 - 0.9153399365i\)
\(L(\frac12)\) \(\approx\) \(-0.4332834622 - 0.9153399365i\)
\(L(1)\) \(\approx\) \(0.5768943930 - 0.8782107248i\)
\(L(1)\) \(\approx\) \(0.5768943930 - 0.8782107248i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
67 \( 1 \)
good2 \( 1 + (0.786 - 0.618i)T \)
3 \( 1 + (-0.327 - 0.945i)T \)
5 \( 1 + (-0.888 - 0.458i)T \)
11 \( 1 + (-0.0475 - 0.998i)T \)
13 \( 1 + (0.415 - 0.909i)T \)
17 \( 1 + (-0.723 + 0.690i)T \)
19 \( 1 + (-0.928 + 0.371i)T \)
23 \( 1 + (0.981 + 0.189i)T \)
29 \( 1 + T \)
31 \( 1 + (-0.995 - 0.0950i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.959 - 0.281i)T \)
43 \( 1 + (0.959 + 0.281i)T \)
47 \( 1 + (-0.981 - 0.189i)T \)
53 \( 1 + (-0.235 + 0.971i)T \)
59 \( 1 + (0.995 + 0.0950i)T \)
61 \( 1 + (0.0475 - 0.998i)T \)
71 \( 1 + (-0.959 + 0.281i)T \)
73 \( 1 + (0.888 - 0.458i)T \)
79 \( 1 + (0.995 - 0.0950i)T \)
83 \( 1 + (-0.841 + 0.540i)T \)
89 \( 1 + (0.327 - 0.945i)T \)
97 \( 1 + T \)
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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

−24.02162084409332908490145051325, −23.42270481211423830073002690206, −22.72758984603872636543055259581, −22.13385437678431766619676431856, −21.15345758264135106161097562572, −20.479000012871412637954622056, −19.49537135867831159259437741521, −18.154359405226384949955357649067, −17.278264661024528001403167983, −16.33952366754499523112048346356, −15.69206401759370798695663676232, −14.98981787659671164319822167950, −14.39204108772302836513254492734, −13.1560601094314962522154075024, −12.04118528076595617751446109757, −11.40355496709973644042371004159, −10.57649973028074169586168893226, −9.17376750985376476708661734487, −8.356737650158309181015243444997, −6.959057529069724691369837242891, −6.56866214788329861161457112192, −4.991982455233505441953965704113, −4.46370561720536734916699541790, −3.59503423835886476145503052843, −2.49265926238457751715634229045, 0.445527566084467742134069884530, 1.57529255551074682254142652661, 2.92877510137459194682409779877, 3.8770246912107743855915965456, 5.10424459943903742488914924725, 5.95342802253562889774619960616, 6.89694105560925483028559269618, 8.131072573092473733183028017973, 8.8638986951835072043769812112, 10.75526821864642897525054913647, 11.01948012467795455197770317359, 12.10485073942929573649118067164, 12.83644718609711075287482614341, 13.333427442381250004606836056206, 14.467585931261530234403084414748, 15.4215731151426852523264133041, 16.30671077973866249424547467852, 17.38515757254041392733381733209, 18.52040900604154409219720486115, 19.29067830647365719925313764351, 19.74631273787971050053887173869, 20.722265378770671496533013463601, 21.687836785388160330809003422593, 22.69094328669199449305432298840, 23.34482315121671528744467756350

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