Properties

Label 1-75-75.38-r0-0-0
Degree $1$
Conductor $75$
Sign $0.904 - 0.425i$
Analytic cond. $0.348298$
Root an. cond. $0.348298$
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  + (−0.951 + 0.309i)2-s + (0.809 − 0.587i)4-s i·7-s + (−0.587 + 0.809i)8-s + (−0.309 − 0.951i)11-s + (0.951 + 0.309i)13-s + (0.309 + 0.951i)14-s + (0.309 − 0.951i)16-s + (0.587 − 0.809i)17-s + (0.809 + 0.587i)19-s + (0.587 + 0.809i)22-s + (0.951 − 0.309i)23-s − 26-s + (−0.587 − 0.809i)28-s + (−0.809 + 0.587i)29-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)2-s + (0.809 − 0.587i)4-s i·7-s + (−0.587 + 0.809i)8-s + (−0.309 − 0.951i)11-s + (0.951 + 0.309i)13-s + (0.309 + 0.951i)14-s + (0.309 − 0.951i)16-s + (0.587 − 0.809i)17-s + (0.809 + 0.587i)19-s + (0.587 + 0.809i)22-s + (0.951 − 0.309i)23-s − 26-s + (−0.587 − 0.809i)28-s + (−0.809 + 0.587i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.904 - 0.425i$
Analytic conductor: \(0.348298\)
Root analytic conductor: \(0.348298\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 75,\ (0:\ ),\ 0.904 - 0.425i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6277125971 - 0.1403103891i\)
\(L(\frac12)\) \(\approx\) \(0.6277125971 - 0.1403103891i\)
\(L(1)\) \(\approx\) \(0.7168041099 - 0.05098787238i\)
\(L(1)\) \(\approx\) \(0.7168041099 - 0.05098787238i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (0.951 - 0.309i)T \)
7 \( 1 + iT \)
11 \( 1 + (0.309 + 0.951i)T \)
13 \( 1 + (-0.951 - 0.309i)T \)
17 \( 1 + (-0.587 + 0.809i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (-0.951 + 0.309i)T \)
29 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (0.951 + 0.309i)T \)
41 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.587 + 0.809i)T \)
53 \( 1 + (-0.587 - 0.809i)T \)
59 \( 1 + (-0.309 + 0.951i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 + (0.587 - 0.809i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (0.951 - 0.309i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (0.587 - 0.809i)T \)
89 \( 1 + (-0.309 - 0.951i)T \)
97 \( 1 + (-0.587 - 0.809i)T \)
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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

−31.0553536108510841008071089127, −30.465250648878010210709889539131, −28.97239338633068427614034434417, −28.22941264950983322824658547140, −27.44451731178880308994188774502, −25.93384357700919609892802668439, −25.42388446865553141440469397028, −24.158249444124225298387204287070, −22.60742946356364273408425988785, −21.32277816518017225813028535579, −20.46273145924013282157486846713, −19.16020434786480255702686625443, −18.259308266382553835948078379189, −17.317096836522730141952697342806, −15.86700956432759316424452197933, −15.07393014027136366369534241268, −13.011617702537495228915043069393, −11.97313773242222682970012585046, −10.753085653395982439910299909212, −9.47869641732391669583999241409, −8.440709690074058481554716518056, −7.12367568875183454844086613756, −5.5491209739343279232784261584, −3.32510029966825084030377005415, −1.78699604115173935807243032772, 1.15503521824423799031228555911, 3.34155928823966303647092930774, 5.49023834111015198147940543087, 6.91612392450805078197485244200, 8.02094741861954636953885262277, 9.31030876345000101284605446228, 10.599058609955312612626892558014, 11.48257501192465130382818618969, 13.43528145130539834909676815283, 14.56375420033985332782888569912, 16.20061750278756078495258895032, 16.64645813799725116509384437254, 18.143706475802455524262351759221, 18.94686521063917165609653315398, 20.27501987926134332460900890389, 21.05332376861695878671878168204, 22.94826163469359129124202868127, 23.89616474911817426022360301560, 24.97518487408460141705291959395, 26.19714466228303504981323284137, 26.8792786852976670057033359143, 27.92979415695613271567661265264, 29.21532370462542981322815391729, 29.81257749884403798385005462226, 31.34722132849131395333040354214

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