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Properties

Label 1-175-175.122-r0-0-0
Degree $1$
Conductor $175$
Sign $-0.784 + 0.620i$
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  + (0.406 + 0.913i)2-s + (0.743 + 0.669i)3-s + (−0.669 + 0.743i)4-s + (−0.309 + 0.951i)6-s + (−0.951 − 0.309i)8-s + (0.104 + 0.994i)9-s + (−0.104 + 0.994i)11-s + (−0.994 + 0.104i)12-s + (−0.587 − 0.809i)13-s + (−0.104 − 0.994i)16-s + (0.207 + 0.978i)17-s + (−0.866 + 0.5i)18-s + (0.669 + 0.743i)19-s + (−0.951 + 0.309i)22-s + (−0.406 − 0.913i)23-s + (−0.5 − 0.866i)24-s + ⋯
L(s)  = 1  + (0.406 + 0.913i)2-s + (0.743 + 0.669i)3-s + (−0.669 + 0.743i)4-s + (−0.309 + 0.951i)6-s + (−0.951 − 0.309i)8-s + (0.104 + 0.994i)9-s + (−0.104 + 0.994i)11-s + (−0.994 + 0.104i)12-s + (−0.587 − 0.809i)13-s + (−0.104 − 0.994i)16-s + (0.207 + 0.978i)17-s + (−0.866 + 0.5i)18-s + (0.669 + 0.743i)19-s + (−0.951 + 0.309i)22-s + (−0.406 − 0.913i)23-s + (−0.5 − 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.784 + 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.784 + 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $-0.784 + 0.620i$
Analytic conductor: \(0.812696\)
Root analytic conductor: \(0.812696\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 175,\ (0:\ ),\ -0.784 + 0.620i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5018319137 + 1.443590038i\)
\(L(\frac12)\) \(\approx\) \(0.5018319137 + 1.443590038i\)
\(L(1)\) \(\approx\) \(0.9564784235 + 1.028139251i\)
\(L(1)\) \(\approx\) \(0.9564784235 + 1.028139251i\)

Euler product

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

−26.96199260441686194835729382513, −26.35364940725587109694652929503, −24.88498589561830011853196514030, −24.09725467921566943061161335757, −23.32397921893335588738463754105, −21.97949834581536349402988861129, −21.23689244765734721051842303572, −20.14300731648942444094795435013, −19.438473227121206716823274621652, −18.60174920826528513307642718483, −17.74870110087998639554765768576, −16.09045315765093464220807198244, −14.72385701471948991920645549678, −13.886600585617522920104726532571, −13.26091190234461084917439355047, −12.00561980233881122675497714233, −11.30425068614709929936853431421, −9.694332468815566466501484799, −8.99362425964340450130853973093, −7.646395058265128981191285165882, −6.289951344987142702330991392645, −4.87570705395865869696403398162, −3.41254218221054748822748077979, −2.53514227650722914181929187406, −1.10245994958893801736044429643, 2.50149578376231151534663286965, 3.841875676666467059757756879373, 4.772347736806875681081767766300, 5.97530061508104814749637756796, 7.556929088577248351088013890092, 8.16387318197576787595319557950, 9.501911070718363458358808253168, 10.31258788690831324579986761192, 12.19698608161871062400369791709, 13.11981257135350424970381405453, 14.31642947614434926889625332729, 14.97762048401200439801767876169, 15.764861681914546282620172062814, 16.83797962924332823564909802466, 17.780854575255946773465223859449, 19.07865928629198015987160224831, 20.3409729975663003828505565, 21.09461208307185345331648773420, 22.29600954418920462230651421522, 22.81476296059806044867057257259, 24.23049324217496725210957580655, 25.0421493011270394251049676610, 25.824448962340595394661777968287, 26.64279801288221529147600770582, 27.466738581511158801886348768726

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