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Properties

Label 1-605-605.87-r0-0-0
Degree $1$
Conductor $605$
Sign $0.994 - 0.108i$
Analytic cond. $2.80960$
Root an. cond. $2.80960$
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  + (0.281 + 0.959i)2-s − i·3-s + (−0.841 + 0.540i)4-s + (0.959 − 0.281i)6-s + (0.909 − 0.415i)7-s + (−0.755 − 0.654i)8-s − 9-s + (0.540 + 0.841i)12-s + (0.540 + 0.841i)13-s + (0.654 + 0.755i)14-s + (0.415 − 0.909i)16-s + (0.989 + 0.142i)17-s + (−0.281 − 0.959i)18-s + (−0.142 − 0.989i)19-s + (−0.415 − 0.909i)21-s + ⋯
L(s)  = 1  + (0.281 + 0.959i)2-s − i·3-s + (−0.841 + 0.540i)4-s + (0.959 − 0.281i)6-s + (0.909 − 0.415i)7-s + (−0.755 − 0.654i)8-s − 9-s + (0.540 + 0.841i)12-s + (0.540 + 0.841i)13-s + (0.654 + 0.755i)14-s + (0.415 − 0.909i)16-s + (0.989 + 0.142i)17-s + (−0.281 − 0.959i)18-s + (−0.142 − 0.989i)19-s + (−0.415 − 0.909i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.994 - 0.108i$
Analytic conductor: \(2.80960\)
Root analytic conductor: \(2.80960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (87, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 605,\ (0:\ ),\ 0.994 - 0.108i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.539112075 - 0.08357638448i\)
\(L(\frac12)\) \(\approx\) \(1.539112075 - 0.08357638448i\)
\(L(1)\) \(\approx\) \(1.204523646 + 0.1205641536i\)
\(L(1)\) \(\approx\) \(1.204523646 + 0.1205641536i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.281 + 0.959i)T \)
3 \( 1 - iT \)
7 \( 1 + (0.909 - 0.415i)T \)
13 \( 1 + (0.540 + 0.841i)T \)
17 \( 1 + (0.989 + 0.142i)T \)
19 \( 1 + (-0.142 - 0.989i)T \)
23 \( 1 + (-0.909 - 0.415i)T \)
29 \( 1 + (-0.142 - 0.989i)T \)
31 \( 1 + (0.841 + 0.540i)T \)
37 \( 1 + (-0.540 + 0.841i)T \)
41 \( 1 + (0.959 - 0.281i)T \)
43 \( 1 + (-0.755 - 0.654i)T \)
47 \( 1 + (0.281 - 0.959i)T \)
53 \( 1 + (0.909 - 0.415i)T \)
59 \( 1 + (0.959 + 0.281i)T \)
61 \( 1 + (0.959 + 0.281i)T \)
67 \( 1 + (-0.281 - 0.959i)T \)
71 \( 1 + (-0.142 - 0.989i)T \)
73 \( 1 + (0.909 + 0.415i)T \)
79 \( 1 + (-0.654 - 0.755i)T \)
83 \( 1 + (-0.909 + 0.415i)T \)
89 \( 1 + (0.142 - 0.989i)T \)
97 \( 1 + (-0.755 - 0.654i)T \)
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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

−22.88756491878187747929774382003, −22.08022897237839081836254043160, −21.25876020070401455833368984914, −20.79829388579602869565388049686, −20.13159303789825190941233112740, −19.09456331861600278474640315209, −18.12953746047726077192147725776, −17.50254985543237683736051653083, −16.32853771493711610808242689794, −15.37354168243335915129935736528, −14.50701572836035429025760296499, −14.06799287323660728986456912070, −12.748440269043581354916006142965, −11.87688490345869525363040961540, −11.15309545070682034483258756236, −10.33360051814429164964163853004, −9.67482907284654532444058092601, −8.56973413065885813945982634716, −7.960582965710826127793192323992, −5.7913215668225263223595154501, −5.43838262375898381762708176519, −4.30093690907373427899313318493, −3.5194209882369268402760823321, −2.51027952031062742764262277266, −1.224727887909587703293572563611, 0.85630681292329152399110347450, 2.17995881778751419884157224303, 3.63017303257675295001747881546, 4.681345937363607989475067298378, 5.68079389536853274679204368480, 6.60643807521380761190351869386, 7.328252188634092319337274036974, 8.22393052821575677753068507119, 8.76104603824627873291694402313, 10.15615632513818744884654927356, 11.54311813977512910831723673374, 12.08705338385453255478353059433, 13.27778816240166391158762409648, 13.86372759450775962267160500517, 14.46225237261786628913430929728, 15.435551367060709836052823335957, 16.5855160937328926433542793575, 17.20899227407818684831024895825, 17.976108045314442520381030290606, 18.651070375643874235437216513219, 19.53871634721822076097018204227, 20.7547835272296268924163818892, 21.45724346104565267158815448485, 22.602242524487225804039063317294, 23.36167401340573211609821206637

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