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Properties

Label 1-75-75.14-r1-0-0
Degree $1$
Conductor $75$
Sign $-0.425 + 0.904i$
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  + (0.309 − 0.951i)2-s + (−0.809 − 0.587i)4-s − 7-s + (−0.809 + 0.587i)8-s + (−0.309 + 0.951i)11-s + (−0.309 − 0.951i)13-s + (−0.309 + 0.951i)14-s + (0.309 + 0.951i)16-s + (−0.809 + 0.587i)17-s + (−0.809 + 0.587i)19-s + (0.809 + 0.587i)22-s + (0.309 − 0.951i)23-s − 26-s + (0.809 + 0.587i)28-s + (0.809 + 0.587i)29-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)2-s + (−0.809 − 0.587i)4-s − 7-s + (−0.809 + 0.587i)8-s + (−0.309 + 0.951i)11-s + (−0.309 − 0.951i)13-s + (−0.309 + 0.951i)14-s + (0.309 + 0.951i)16-s + (−0.809 + 0.587i)17-s + (−0.809 + 0.587i)19-s + (0.809 + 0.587i)22-s + (0.309 − 0.951i)23-s − 26-s + (0.809 + 0.587i)28-s + (0.809 + 0.587i)29-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.425 + 0.904i$
Analytic conductor: \(8.05986\)
Root analytic conductor: \(8.05986\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 75,\ (1:\ ),\ -0.425 + 0.904i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04420616673 - 0.06965777263i\)
\(L(\frac12)\) \(\approx\) \(-0.04420616673 - 0.06965777263i\)
\(L(1)\) \(\approx\) \(0.6357777996 - 0.3495218031i\)
\(L(1)\) \(\approx\) \(0.6357777996 - 0.3495218031i\)

Euler product

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

−31.913920914084507512144951120689, −31.25889403861981853241173415455, −29.748695684589697493594639298151, −28.71733134598476622758510560636, −27.17711782242439620320103706788, −26.31332728536689014915398591631, −25.4147515720087616866961495343, −24.220207542718535533050582807308, −23.3624945114136360745236574117, −22.15492722365398055084108309145, −21.37765446822366796281369381270, −19.53427970298240066701174836940, −18.52680915430992150165182765777, −17.103074852765470190002252635421, −16.1967993687067881097940946607, −15.25190626869013423332180504305, −13.7743420398259030772506897242, −13.061382823037689603893063710, −11.52462731851607203037641272075, −9.650390495473313599727607380685, −8.60660878905825794819903096350, −7.04333535542943088126746612175, −6.09694482112194161064627066633, −4.561584291274186832023498064691, −3.05447251532235183969810889736, 0.034781810516460489136878182495, 2.182683380260347232114542670249, 3.59187693176354839218970107253, 5.05376889430521604329438206235, 6.60456811334367194005075113597, 8.55699999068663908463874167931, 9.938993230513482493878735029520, 10.719379852575104641996969188952, 12.56187923476548048403030568827, 12.83706008157947641544221726297, 14.48364794039986402532465117774, 15.596930751746574346021973010962, 17.29462376070722159238064850937, 18.4357033074746015941940499634, 19.63391163268307454259972031309, 20.34967719885340706729766260319, 21.69246374143386805409167345286, 22.6781705935593212532115021885, 23.42447629278815933291668625798, 25.00817064561824613083744452112, 26.23143522492909115675111228278, 27.4477017023164689029032731087, 28.517708018997239150427243032111, 29.29002048452261153870715910021, 30.38342996382534435769374429951

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