Properties

Label 1-175-175.138-r0-0-0
Degree $1$
Conductor $175$
Sign $0.445 + 0.895i$
Analytic cond. $0.812696$
Root an. cond. $0.812696$
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  + (−0.743 − 0.669i)2-s + (−0.994 + 0.104i)3-s + (0.104 + 0.994i)4-s + (0.809 + 0.587i)6-s + (0.587 − 0.809i)8-s + (0.978 − 0.207i)9-s + (−0.978 − 0.207i)11-s + (−0.207 − 0.978i)12-s + (−0.951 − 0.309i)13-s + (−0.978 + 0.207i)16-s + (0.406 + 0.913i)17-s + (−0.866 − 0.5i)18-s + (−0.104 + 0.994i)19-s + (0.587 + 0.809i)22-s + (0.743 + 0.669i)23-s + (−0.5 + 0.866i)24-s + ⋯
L(s)  = 1  + (−0.743 − 0.669i)2-s + (−0.994 + 0.104i)3-s + (0.104 + 0.994i)4-s + (0.809 + 0.587i)6-s + (0.587 − 0.809i)8-s + (0.978 − 0.207i)9-s + (−0.978 − 0.207i)11-s + (−0.207 − 0.978i)12-s + (−0.951 − 0.309i)13-s + (−0.978 + 0.207i)16-s + (0.406 + 0.913i)17-s + (−0.866 − 0.5i)18-s + (−0.104 + 0.994i)19-s + (0.587 + 0.809i)22-s + (0.743 + 0.669i)23-s + (−0.5 + 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.445 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.445 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.445 + 0.895i$
Analytic conductor: \(0.812696\)
Root analytic conductor: \(0.812696\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (138, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 175,\ (0:\ ),\ 0.445 + 0.895i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3067296237 + 0.1899497710i\)
\(L(\frac12)\) \(\approx\) \(0.3067296237 + 0.1899497710i\)
\(L(1)\) \(\approx\) \(0.4755559150 + 0.01501079515i\)
\(L(1)\) \(\approx\) \(0.4755559150 + 0.01501079515i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.743 - 0.669i)T \)
3 \( 1 + (-0.994 + 0.104i)T \)
11 \( 1 + (-0.978 - 0.207i)T \)
13 \( 1 + (-0.951 - 0.309i)T \)
17 \( 1 + (0.406 + 0.913i)T \)
19 \( 1 + (-0.104 + 0.994i)T \)
23 \( 1 + (0.743 + 0.669i)T \)
29 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (-0.913 + 0.406i)T \)
37 \( 1 + (0.207 + 0.978i)T \)
41 \( 1 + (-0.309 + 0.951i)T \)
43 \( 1 + iT \)
47 \( 1 + (-0.406 + 0.913i)T \)
53 \( 1 + (0.994 - 0.104i)T \)
59 \( 1 + (0.669 + 0.743i)T \)
61 \( 1 + (-0.669 + 0.743i)T \)
67 \( 1 + (-0.406 - 0.913i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (-0.207 + 0.978i)T \)
79 \( 1 + (-0.913 - 0.406i)T \)
83 \( 1 + (-0.587 + 0.809i)T \)
89 \( 1 + (0.669 - 0.743i)T \)
97 \( 1 + (-0.587 - 0.809i)T \)
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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

−27.22534082131678535326231047353, −26.498858185785677208968446112921, −25.322843095541655419421399441402, −24.33136254583339749008737382406, −23.603762187996360658859258975733, −22.77181874637080435810677789572, −21.6349741810522292394735383615, −20.34023267295774953096387642347, −19.088370103766340243340205419338, −18.2780129138789156265500616627, −17.476681834357311352345222810249, −16.553343484370995173405559310830, −15.77922412118321518605448333611, −14.71949210050279437838873461001, −13.349313098946007476558138376548, −12.09877824210947254547826075568, −10.92646600573950232027884506584, −10.11775641079642205488349537450, −8.99156513593260868599286449539, −7.43885651830127458819160951535, −6.89686312323490525413222250016, −5.431445663510647842423268257674, −4.79302943591295084534240857387, −2.2793338255288206203079614697, −0.44312884633739637845855155136, 1.39840650279236173281638855105, 3.03375131263195764569041214237, 4.51730786940635794139665925600, 5.8122589763346258078826476797, 7.25803893873458065180534125756, 8.21976416848562029927775066254, 9.82414503132680088743886583081, 10.39513634907500374597058581164, 11.436985058233374260216744125914, 12.42548181902583971042252884728, 13.15889125056483156166394130117, 15.037072468642195184312553241341, 16.23222601702683337659379896868, 17.00153385128223348417088363652, 17.87120664214459287585569180802, 18.743169300768324155388318321259, 19.6836236447599594465671453777, 21.07264317804708520678177433170, 21.55651025068121055856131931605, 22.673673995248028572815688941563, 23.61825710324789273477475393870, 24.82221763370181810025530325018, 25.961672817431794373576863069896, 27.06982386648184442925526519733, 27.535216293742584759101068430207

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