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Properties

Label 1-35-35.4-r0-0-0
Degree $1$
Conductor $35$
Sign $0.832 - 0.553i$
Analytic cond. $0.162539$
Root an. cond. $0.162539$
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  + (0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + 6-s − 8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)11-s + (0.5 − 0.866i)12-s − 13-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (0.5 + 0.866i)18-s + (−0.5 + 0.866i)19-s − 22-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)24-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + 6-s − 8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)11-s + (0.5 − 0.866i)12-s − 13-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (0.5 + 0.866i)18-s + (−0.5 + 0.866i)19-s − 22-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.832 - 0.553i$
Analytic conductor: \(0.162539\)
Root analytic conductor: \(0.162539\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 35,\ (0:\ ),\ 0.832 - 0.553i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9621136901 - 0.2907644797i\)
\(L(\frac12)\) \(\approx\) \(0.9621136901 - 0.2907644797i\)
\(L(1)\) \(\approx\) \(1.182550899 - 0.2880453366i\)
\(L(1)\) \(\approx\) \(1.182550899 - 0.2880453366i\)

Euler product

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

−36.08010718347583973968305717344, −34.78630270406825048659816198443, −33.85168316588980005729575331500, −32.30928966522983925166123340399, −31.374976177374825121337147647162, −30.41169231701160649511850213476, −29.16113552711797203041549793347, −27.20226581808885622263080204850, −25.84798696317896843467964966942, −25.06596384401760540316394029942, −23.86647852673818350801122913858, −22.93913122922319865029391755753, −21.32557264710106733534653636906, −19.82147557556496181734857256820, −18.23251300007088022367460911928, −17.22078314988795062833112294854, −15.451543861962476732968840903988, −14.34263834869501845261986182652, −13.103129913852945798305016985618, −12.04355413900294731187371566602, −9.35121393364928968506912291420, −7.77099119303511624665327929642, −6.82720838437033716297025886149, −4.966591444039001890276103830016, −2.805415288560253200155041975893, 2.61073590462750611267003234178, 4.091941906320124082230531473735, 5.59516565956985674656901314720, 8.40858836724316330810239083309, 9.918733084605706971412125539, 10.92057126519500693615094455589, 12.61621287361943043574665698501, 14.12060020806093949942599269037, 15.08199576272651307855651437445, 16.725315590376243287885299404881, 18.79205242684371415669800391450, 19.839655158924446943536702732553, 21.12535891659633864055846180971, 21.83668478533503203207788674412, 23.20549606148024611708085954174, 24.71913309525814152096509173139, 26.47929341772145333016249793723, 27.40500602157178847660448058408, 28.65241918871529937068080549835, 29.88170457154007418886137650784, 31.29058485474928322871332656331, 32.03398042913925170336934093007, 33.051160230393815169485762796088, 34.41397088857799204348606602112, 36.54697848660964409628709153500

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