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Properties

Label 1-75-75.29-r1-0-0
Degree $1$
Conductor $75$
Sign $-0.0627 - 0.998i$
Analytic cond. $8.05986$
Root an. cond. $8.05986$
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  + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s − 7-s + (0.309 − 0.951i)8-s + (0.809 + 0.587i)11-s + (0.809 − 0.587i)13-s + (0.809 + 0.587i)14-s + (−0.809 + 0.587i)16-s + (0.309 − 0.951i)17-s + (0.309 − 0.951i)19-s + (−0.309 − 0.951i)22-s + (−0.809 − 0.587i)23-s − 26-s + (−0.309 − 0.951i)28-s + (−0.309 − 0.951i)29-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s − 7-s + (0.309 − 0.951i)8-s + (0.809 + 0.587i)11-s + (0.809 − 0.587i)13-s + (0.809 + 0.587i)14-s + (−0.809 + 0.587i)16-s + (0.309 − 0.951i)17-s + (0.309 − 0.951i)19-s + (−0.309 − 0.951i)22-s + (−0.809 − 0.587i)23-s − 26-s + (−0.309 − 0.951i)28-s + (−0.309 − 0.951i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0627 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0627 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.0627 - 0.998i$
Analytic conductor: \(8.05986\)
Root analytic conductor: \(8.05986\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 75,\ (1:\ ),\ -0.0627 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6186552539 - 0.6588009319i\)
\(L(\frac12)\) \(\approx\) \(0.6186552539 - 0.6588009319i\)
\(L(1)\) \(\approx\) \(0.6745711995 - 0.2670816318i\)
\(L(1)\) \(\approx\) \(0.6745711995 - 0.2670816318i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (-0.809 - 0.587i)T \)
7 \( 1 - T \)
11 \( 1 + (0.809 + 0.587i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (-0.809 - 0.587i)T \)
29 \( 1 + (-0.309 - 0.951i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (0.809 - 0.587i)T \)
41 \( 1 + (0.809 - 0.587i)T \)
43 \( 1 - T \)
47 \( 1 + (0.309 + 0.951i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.809 - 0.587i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (-0.309 + 0.951i)T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (0.309 - 0.951i)T \)
89 \( 1 + (0.809 + 0.587i)T \)
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

−31.75349440960455470478102006366, −30.077046485710029839327527119767, −29.02790695945098416389827673486, −28.120268693628343308049274426298, −27.007759824986610116150256778486, −25.958008911049505381577090183753, −25.21457389958965754837935692096, −23.96010920999041099860235042911, −22.98045151066318833276914889148, −21.607725978389323125843301682163, −19.99354281581144988121257733504, −19.17122787788890232044546931487, −18.17450471056968502257826957388, −16.68359267747745592251805682454, −16.17446912566509451624708963149, −14.74579422935854980789671760157, −13.5805229634776756424778887777, −11.87500967273793395671203314554, −10.46225495016986009119903782230, −9.34511289589187023609734684701, −8.25206795760349723209219285563, −6.677972549457157845210349042613, −5.83894634426235414710903452448, −3.640475470701649449903715557477, −1.38164771200965176663227890790, 0.647907608515370417496603222912, 2.61090157321330634432956947173, 4.003385295682352927990601534257, 6.307497343845292934227362329571, 7.58869242656452991122921431304, 9.121872837726798025008402633897, 9.92747520086264918832243951586, 11.32603735525613577757663060260, 12.458776819574337630127543410418, 13.58594927417852583037112132432, 15.51289036467063384165577920791, 16.49683436631425155559496634625, 17.69209528184444908282838552265, 18.739107524374311762802986348833, 19.85355271015115075915981507350, 20.61578927793656665064395323585, 22.11648667142112123431858224126, 22.871723897208606614152518445975, 24.7744599278026936073990638669, 25.66400627936200488025477691999, 26.56951873835590387829604047053, 27.87672844179880460903969386440, 28.51520663157403303218353490160, 29.78972909650278053946904323632, 30.45421313028494474932501023914

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