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Properties

Label 1-175-175.128-r1-0-0
Degree $1$
Conductor $175$
Sign $0.309 - 0.950i$
Analytic cond. $18.8063$
Root an. cond. $18.8063$
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  + (0.994 + 0.104i)2-s + (0.207 − 0.978i)3-s + (0.978 + 0.207i)4-s + (0.309 − 0.951i)6-s + (0.951 + 0.309i)8-s + (−0.913 − 0.406i)9-s + (0.913 − 0.406i)11-s + (0.406 − 0.913i)12-s + (−0.587 − 0.809i)13-s + (0.913 + 0.406i)16-s + (0.743 − 0.669i)17-s + (−0.866 − 0.5i)18-s + (0.978 − 0.207i)19-s + (0.951 − 0.309i)22-s + (−0.994 − 0.104i)23-s + (0.5 − 0.866i)24-s + ⋯
L(s)  = 1  + (0.994 + 0.104i)2-s + (0.207 − 0.978i)3-s + (0.978 + 0.207i)4-s + (0.309 − 0.951i)6-s + (0.951 + 0.309i)8-s + (−0.913 − 0.406i)9-s + (0.913 − 0.406i)11-s + (0.406 − 0.913i)12-s + (−0.587 − 0.809i)13-s + (0.913 + 0.406i)16-s + (0.743 − 0.669i)17-s + (−0.866 − 0.5i)18-s + (0.978 − 0.207i)19-s + (0.951 − 0.309i)22-s + (−0.994 − 0.104i)23-s + (0.5 − 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.309 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.309 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.309 - 0.950i$
Analytic conductor: \(18.8063\)
Root analytic conductor: \(18.8063\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 175,\ (1:\ ),\ 0.309 - 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.113930329 - 2.261110535i\)
\(L(\frac12)\) \(\approx\) \(3.113930329 - 2.261110535i\)
\(L(1)\) \(\approx\) \(2.058528579 - 0.7464453301i\)
\(L(1)\) \(\approx\) \(2.058528579 - 0.7464453301i\)

Euler product

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

−27.52059699882776671414498048161, −26.27020202896245831168343693898, −25.47177217260456126384154647036, −24.41670135006961920540093258263, −23.40187979647349385970863044931, −22.16271477123628634306893058500, −21.95031124492721781027327009618, −20.697031979328224534315515459701, −20.04062886089656734373924659610, −19.03124546941551679448268082839, −17.12853025941731578734892223341, −16.415073111931738586153991321682, −15.33855237240042059255198817663, −14.46772488361690054027879338481, −13.842968977210431027432089543155, −12.23660117913084523776840126171, −11.55324914332112303614491768346, −10.25483476287541765197484570391, −9.426763374079732322992957085239, −7.832163629154197173635847124465, −6.44827485846133775504214697811, −5.249697731096097424870863144403, −4.21123465532723162402866762424, −3.32341230507162178248838360010, −1.8168319767267591307449966188, 1.039366340528021762440347184471, 2.53931481893073648340792135976, 3.58070271886222369929289489035, 5.23515564005333217264853844203, 6.237749660336317005685370696395, 7.30137967455115481388973468457, 8.18863891734117602010969126466, 9.84637996099312052051131784469, 11.527177234690754514411846790601, 12.054061323813314344602726207106, 13.15754168510768198815494030862, 14.05038778990666794489849378189, 14.7416927109345466860018494098, 16.080596046890681517747454802834, 17.14544413494849180794363889714, 18.23508814269631890842620671289, 19.59943845503027194529946029360, 20.07103363370927224125723121771, 21.31080341879003789539363863881, 22.51769479719852574550301532163, 23.05025382193886436572237439141, 24.40052846838787265157544851907, 24.68557888278981903279655950857, 25.62726928652367793948194770397, 26.81596171585069665069946707656

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