Properties

Label 1-35-35.3-r0-0-0
Degree $1$
Conductor $35$
Sign $0.413 + 0.910i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.413 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.413 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.413 + 0.910i$
Analytic conductor: \(0.162539\)
Root analytic conductor: \(0.162539\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 35,\ (0:\ ),\ 0.413 + 0.910i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7991983432 + 0.5145033684i\)
\(L(\frac12)\) \(\approx\) \(0.7991983432 + 0.5145033684i\)
\(L(1)\) \(\approx\) \(1.058335931 + 0.4733744821i\)
\(L(1)\) \(\approx\) \(1.058335931 + 0.4733744821i\)

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

−35.80013473603395144869932470468, −34.158330518742277288403798722381, −33.51379303900399491664496161263, −32.0350327755604333546765305005, −30.71671367385168065015716871519, −29.84923286999480506900671721143, −28.60477526369126469121963508977, −27.91872212879256090594703309199, −25.7071536840550789463868600973, −24.10864515671436679354702270075, −23.4376316269784483465833604334, −22.22675845269073007084478239975, −21.11571848579420403991928649107, −19.52193822755942282423909543918, −18.33506531340939685212571873723, −16.71797000479940638888174294937, −15.20299177757423858075662843866, −13.573690714928856642771938895439, −12.42726294034201006771538127779, −11.35423713312006801274194634192, −9.97974127688517494176518911515, −7.24740272572690371684591513095, −5.83395042694476754865003442461, −4.392224696474392375853104380304, −1.969169021137100745218735244449, 3.43481525921369310600501690633, 5.138662754376125234002452406466, 6.19593039359663724126985999563, 8.03553049727166256414221079787, 10.33870006154565848158924862132, 11.72876029289555446369806732339, 12.97760753792570457240425600923, 14.62692708812760852913901544285, 15.94113780529554755218316419443, 16.81027988209426194463775545133, 18.271195512051699914532146571234, 20.55777022636353710996272154172, 21.59312202463794910859267667105, 22.71506978607732521960258885206, 23.62416837487569539975816556832, 24.912832447440480017061203137664, 26.40286115223966556390009930542, 27.60985463714245214892458399196, 29.21732571396782478724090916253, 30.08133720772457709176164179164, 31.84906682296229395054317923286, 32.524606221947250608406599823684, 33.93429512782188324977923086595, 34.4802257503679678150429883356, 35.749660366071901960504264976776

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