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Properties

Label 1-35-35.2-r1-0-0
Degree $1$
Conductor $35$
Sign $0.908 - 0.417i$
Analytic cond. $3.76127$
Root an. cond. $3.76127$
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  + (0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (0.5 − 0.866i)4-s + 6-s − i·8-s + (0.5 + 0.866i)9-s + (−0.5 + 0.866i)11-s + (0.866 − 0.5i)12-s − i·13-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s + (0.866 + 0.5i)18-s + (0.5 + 0.866i)19-s + i·22-s + (−0.866 + 0.5i)23-s + (0.5 − 0.866i)24-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (0.5 − 0.866i)4-s + 6-s − i·8-s + (0.5 + 0.866i)9-s + (−0.5 + 0.866i)11-s + (0.866 − 0.5i)12-s − i·13-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s + (0.866 + 0.5i)18-s + (0.5 + 0.866i)19-s + i·22-s + (−0.866 + 0.5i)23-s + (0.5 − 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.908 - 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.908 - 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.908 - 0.417i$
Analytic conductor: \(3.76127\)
Root analytic conductor: \(3.76127\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 35,\ (1:\ ),\ 0.908 - 0.417i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.718111101 - 0.5939212068i\)
\(L(\frac12)\) \(\approx\) \(2.718111101 - 0.5939212068i\)
\(L(1)\) \(\approx\) \(2.022687173 - 0.3439784556i\)
\(L(1)\) \(\approx\) \(2.022687173 - 0.3439784556i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 - iT \)
17 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 - T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 + T \)
43 \( 1 - iT \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (0.866 + 0.5i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + T \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 - iT \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + iT \)
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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

−35.59910556888870191104038261084, −34.5491197196213397853178449292, −33.139790922880960360837633984477, −31.98429806201945131277114662520, −31.14316437994159622782128062278, −30.12097472286172158098414098320, −28.91438142656687972759492667454, −26.53942157582756157533475900835, −25.94738826735708365552515402428, −24.31453507290520542980915996304, −23.96053750711599078301222154631, −22.10785158455013824657689698595, −20.9722004665645462449841208562, −19.6913951972389108501187299340, −18.16097632199931193431851527847, −16.40798799212429862133596127345, −15.09673006095517454118607126277, −13.870216007044681308131658372130, −12.99469199329751140900192168461, −11.40305618408171642726960277184, −8.9326723563768611725471447826, −7.59617739475376023513821328415, −6.195058614860654066255917608380, −4.10981455216769926587429169531, −2.45955293864876159220403598747, 2.25482763699762490408729632759, 3.800184639831668899720305117076, 5.30810329549512608468832877963, 7.53940282731089718088345180747, 9.56053378114159156516744300472, 10.71454759968517399891237260839, 12.54940421952370673502148455321, 13.75107163835168794846892456630, 15.003608723005802107184334105923, 15.93436159450332709011586603405, 18.302937061750072741358329909129, 19.954979484248753356949264425323, 20.51981840154720851774899142914, 21.867376268165700036905272143251, 22.96795193488978953650872254987, 24.567079380133396770061055546504, 25.59616376233569921601209715488, 27.188152775860030134872747315948, 28.40474534221674878824998352695, 29.879416518472621948159014955233, 31.00485230188198038257288282362, 31.80334912458809387038562252908, 32.96917005755833550252605032144, 33.84273960730530060122891935750, 35.89377509879203932391478691188

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