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Properties

Label 1-175-175.86-r0-0-0
Degree $1$
Conductor $175$
Sign $0.992 - 0.124i$
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  + (−0.978 + 0.207i)2-s + (0.913 − 0.406i)3-s + (0.913 − 0.406i)4-s + (−0.809 + 0.587i)6-s + (−0.809 + 0.587i)8-s + (0.669 − 0.743i)9-s + (0.669 + 0.743i)11-s + (0.669 − 0.743i)12-s + (0.309 + 0.951i)13-s + (0.669 − 0.743i)16-s + (−0.104 − 0.994i)17-s + (−0.5 + 0.866i)18-s + (0.913 + 0.406i)19-s + (−0.809 − 0.587i)22-s + (−0.978 + 0.207i)23-s + (−0.5 + 0.866i)24-s + ⋯
L(s)  = 1  + (−0.978 + 0.207i)2-s + (0.913 − 0.406i)3-s + (0.913 − 0.406i)4-s + (−0.809 + 0.587i)6-s + (−0.809 + 0.587i)8-s + (0.669 − 0.743i)9-s + (0.669 + 0.743i)11-s + (0.669 − 0.743i)12-s + (0.309 + 0.951i)13-s + (0.669 − 0.743i)16-s + (−0.104 − 0.994i)17-s + (−0.5 + 0.866i)18-s + (0.913 + 0.406i)19-s + (−0.809 − 0.587i)22-s + (−0.978 + 0.207i)23-s + (−0.5 + 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.992 - 0.124i$
Analytic conductor: \(0.812696\)
Root analytic conductor: \(0.812696\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (86, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 175,\ (0:\ ),\ 0.992 - 0.124i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.060685290 - 0.06644336577i\)
\(L(\frac12)\) \(\approx\) \(1.060685290 - 0.06644336577i\)
\(L(1)\) \(\approx\) \(0.9830130804 - 0.03353680023i\)
\(L(1)\) \(\approx\) \(0.9830130804 - 0.03353680023i\)

Euler product

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

−27.44440395488474311831949216885, −26.47708915826396638014903969376, −25.81817633437426227842150458520, −24.840179316595867792515857236, −24.1183251168755305792742246435, −22.205845638267358700522319684620, −21.49462303033989664367703137581, −20.292476359452610785758789510487, −19.83005796959670197981731604806, −18.8251145590750107164582752662, −17.83566604743282140463564438834, −16.62136831573418287715924374912, −15.7705431106538978145161674248, −14.82690357214438450873909879352, −13.608743355107316680625732963045, −12.371897858029345895723839229598, −11.00223550953749861190393532673, −10.17921641373240634795272073337, −9.050402005512988232775172819920, −8.338241280219475207266566698392, −7.30816711568299867121502783072, −5.8372188996147506151371403477, −3.85036580462966733702923467269, −2.91603183652859106386719028118, −1.43672822489629683175790990734, 1.41155224037397409159044096100, 2.49817906852989747341746393090, 4.07588573538949873125662158932, 6.083928840464335881865990677350, 7.204754663751363594511936787514, 7.91982694995237897332974857020, 9.42112547795727196319258704414, 9.51925129455416682822239047681, 11.34740406830161472931087497081, 12.24788295163309934534519503241, 13.7967419083117784542677714873, 14.6126383437551843544930183508, 15.67939811640775131106080388890, 16.65733909965744018706280482888, 17.9639842530175098057358204554, 18.56506030043022353549414378156, 19.632279700893460306100469029803, 20.31835425735369052405353973541, 21.174989828200476586124442703851, 22.76579680594947651670208617253, 24.11059037061448255130029882581, 24.707139596395597742286087615376, 25.66172019188522874977986616394, 26.33019009002838415202799063877, 27.241500574982140998632377821892

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