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Properties

Label 1-35-35.33-r0-0-0
Degree $1$
Conductor $35$
Sign $0.629 + 0.777i$
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  + (−0.866 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.629 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.629 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.629 + 0.777i$
Analytic conductor: \(0.162539\)
Root analytic conductor: \(0.162539\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 35,\ (0:\ ),\ 0.629 + 0.777i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6116804491 + 0.2918893856i\)
\(L(\frac12)\) \(\approx\) \(0.6116804491 + 0.2918893856i\)
\(L(1)\) \(\approx\) \(0.7926978904 + 0.2711572794i\)
\(L(1)\) \(\approx\) \(0.7926978904 + 0.2711572794i\)

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

−35.96270086253122801640028356519, −35.09296740751776314898562212025, −33.65518575505444040652958940308, −31.78127560345811002225780429535, −30.87003465199653454561197775295, −29.63444560153303177754614178276, −28.732147441629365788061390699132, −27.0177541474684750262038974377, −26.23888240716562225938201244508, −25.057285989709740448874949177, −23.854431123206774184264590205150, −21.62390064472706299153577719837, −20.65134006054562680935884971093, −19.28048453256467808208508314869, −18.63918325735076744651007300612, −17.10377650840336635045748007481, −15.560659752568304742491375763210, −13.776630047602871880605957776918, −12.472599290410091491840568551538, −10.89544887421104472732461505599, −9.22161594293743291918430494614, −8.21038992951406096409383322199, −6.74733222095866922720308575096, −3.57358406320097593820261441512, −1.90826950242874198743818746701, 2.46964557874449569054137389222, 4.93285715404158198128287923772, 7.12060010875330831846743966380, 8.42647097308994696046667306293, 9.66593677165573394486184262328, 10.854025857985630462267460608935, 13.25844629396880483381306945062, 14.977361652386276259721987325088, 15.60366176059770128168371641262, 17.21420766694968521306612889552, 18.566989747968578274225557523680, 19.92702506936952011911507609472, 20.738334712893500248621441580443, 22.63299749653928026841191096058, 24.33717797642689553020890424716, 25.42025122175247016296894445952, 26.313442973567259426520735517494, 27.42257289457287173816836169136, 28.459085286747930218900475285782, 30.10673886355030755446715922793, 31.568146779162031669015226100211, 32.798583434149184894956405391408, 33.66246136474567803781005315002, 35.09475637831373976587767440636, 36.32077338464839187938891941646

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