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Properties

Label 1-175-175.156-r0-0-0
Degree $1$
Conductor $175$
Sign $0.186 + 0.982i$
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  + (0.913 + 0.406i)2-s + (0.669 + 0.743i)3-s + (0.669 + 0.743i)4-s + (0.309 + 0.951i)6-s + (0.309 + 0.951i)8-s + (−0.104 + 0.994i)9-s + (−0.104 − 0.994i)11-s + (−0.104 + 0.994i)12-s + (−0.809 − 0.587i)13-s + (−0.104 + 0.994i)16-s + (−0.978 − 0.207i)17-s + (−0.5 + 0.866i)18-s + (0.669 − 0.743i)19-s + (0.309 − 0.951i)22-s + (0.913 + 0.406i)23-s + (−0.5 + 0.866i)24-s + ⋯
L(s)  = 1  + (0.913 + 0.406i)2-s + (0.669 + 0.743i)3-s + (0.669 + 0.743i)4-s + (0.309 + 0.951i)6-s + (0.309 + 0.951i)8-s + (−0.104 + 0.994i)9-s + (−0.104 − 0.994i)11-s + (−0.104 + 0.994i)12-s + (−0.809 − 0.587i)13-s + (−0.104 + 0.994i)16-s + (−0.978 − 0.207i)17-s + (−0.5 + 0.866i)18-s + (0.669 − 0.743i)19-s + (0.309 − 0.951i)22-s + (0.913 + 0.406i)23-s + (−0.5 + 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.186 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.186 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.186 + 0.982i$
Analytic conductor: \(0.812696\)
Root analytic conductor: \(0.812696\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (156, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 175,\ (0:\ ),\ 0.186 + 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.724214182 + 1.427183203i\)
\(L(\frac12)\) \(\approx\) \(1.724214182 + 1.427183203i\)
\(L(1)\) \(\approx\) \(1.730972144 + 0.9144977825i\)
\(L(1)\) \(\approx\) \(1.730972144 + 0.9144977825i\)

Euler product

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

−27.22251719966340627763621183128, −26.01772320082470667580256714866, −25.008596044475990158492213022975, −24.339069125851163806557174352937, −23.407713320577184524710375100628, −22.50188100774064079907584289439, −21.36594839033843814111954450315, −20.336408262379719620779062426133, −19.747396761203559272346685787673, −18.74002943382334547807089331579, −17.677609167586493772292410566528, −16.14328077297247309570913225706, −14.82519161386866466903445628252, −14.41372710534778237529570306286, −13.131522770820410609745804738512, −12.507222325333697408557327319760, −11.51816515711024895821575988864, −10.100905278224578617267712629187, −8.99729289064550953116106387267, −7.349332607852746739203598574201, −6.70721048315556307478660966468, −5.16488893783887237915494031583, −3.91836066463791870008498854572, −2.58885419825667096792876057463, −1.63130441763341347159084756603, 2.502389997472929443423803666728, 3.377340909931695231051786178130, 4.6578694566302039090206665962, 5.545003213967157399639985888044, 7.08094867388737655364984528411, 8.165302157987502729768000533881, 9.24538184494304449299172439895, 10.70050448082141552053611528668, 11.61945360306147631709100825298, 13.17296737229100663717960582069, 13.76056958287073810439700340710, 14.92227189494168795115812472970, 15.575599762828475153715547618451, 16.50871960788738228737164754380, 17.55065593662058203495937600701, 19.27833585288688283739033421112, 20.16010444876051403197648183734, 21.05869949435877007888033614775, 22.00825139849316017887333151786, 22.510972813895331168282222707916, 24.01050584514020564648890585833, 24.69951795628989389339771461186, 25.63424662433143021335580795885, 26.64501580036760548547517596346, 27.21285497916599611819083236271

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