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Properties

Label 1-1925-1925.1048-r0-0-0
Degree $1$
Conductor $1925$
Sign $0.607 + 0.794i$
Analytic cond. $8.93966$
Root an. cond. $8.93966$
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  + (0.994 + 0.104i)2-s + (0.866 + 0.5i)3-s + (0.978 + 0.207i)4-s + (0.809 + 0.587i)6-s + (0.951 + 0.309i)8-s + (0.5 + 0.866i)9-s + (0.743 + 0.669i)12-s − i·13-s + (0.913 + 0.406i)16-s + (0.406 + 0.913i)17-s + (0.406 + 0.913i)18-s + (−0.978 + 0.207i)19-s + (−0.207 − 0.978i)23-s + (0.669 + 0.743i)24-s + (0.104 − 0.994i)26-s + i·27-s + ⋯
L(s)  = 1  + (0.994 + 0.104i)2-s + (0.866 + 0.5i)3-s + (0.978 + 0.207i)4-s + (0.809 + 0.587i)6-s + (0.951 + 0.309i)8-s + (0.5 + 0.866i)9-s + (0.743 + 0.669i)12-s − i·13-s + (0.913 + 0.406i)16-s + (0.406 + 0.913i)17-s + (0.406 + 0.913i)18-s + (−0.978 + 0.207i)19-s + (−0.207 − 0.978i)23-s + (0.669 + 0.743i)24-s + (0.104 − 0.994i)26-s + i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.607 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.607 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $0.607 + 0.794i$
Analytic conductor: \(8.93966\)
Root analytic conductor: \(8.93966\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (1048, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1925,\ (0:\ ),\ 0.607 + 0.794i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.186485717 + 2.067980675i\)
\(L(\frac12)\) \(\approx\) \(4.186485717 + 2.067980675i\)
\(L(1)\) \(\approx\) \(2.570125967 + 0.7318249539i\)
\(L(1)\) \(\approx\) \(2.570125967 + 0.7318249539i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.994 + 0.104i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 - iT \)
17 \( 1 + (0.406 + 0.913i)T \)
19 \( 1 + (-0.978 + 0.207i)T \)
23 \( 1 + (-0.207 - 0.978i)T \)
29 \( 1 + (-0.309 - 0.951i)T \)
31 \( 1 + (0.978 + 0.207i)T \)
37 \( 1 + (0.207 + 0.978i)T \)
41 \( 1 + (0.809 + 0.587i)T \)
43 \( 1 + iT \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (0.994 - 0.104i)T \)
59 \( 1 + (0.669 + 0.743i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (-0.994 + 0.104i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (-0.994 + 0.104i)T \)
79 \( 1 + (-0.913 - 0.406i)T \)
83 \( 1 + (-0.587 + 0.809i)T \)
89 \( 1 + (-0.978 + 0.207i)T \)
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

−20.07527019569437039677689210405, −19.18033548032992455195767512387, −18.934611747823621611877954228131, −17.8086829260051518628729003596, −16.884872262877075612844761850221, −15.949464134379938368456175835181, −15.42550401786236640991755918989, −14.417054101925134916940662525892, −14.14753329544598261539685201842, −13.371767110866031963234322209173, −12.68405221970101370021036183030, −11.95962728854802682455834339775, −11.29868420795782599774631373428, −10.27408736746693128335471489255, −9.38225529333827163789540411074, −8.64063333354700209224677169612, −7.494130447723950240599400609654, −7.09669904960735414258118403561, −6.23711581877360161871746453352, −5.32819871075129949323155659791, −4.25359221902843904452296977415, −3.712310072483674946269271675608, −2.667324361840562353287441673429, −2.08403832365226300462608530273, −1.09446598123238058132665595220, 1.37766220785544195859754976075, 2.50049789373104687992216170586, 2.97576773354048710024643511242, 4.10433280960827909571557398241, 4.40964990886227329275126948887, 5.57633855372860877920536732292, 6.23308548470338964950900971013, 7.29429919525445116810042301836, 8.16982120502447176524432449350, 8.524290160083793363537769373378, 10.03396753958627954659169482746, 10.32321054944942265553121633333, 11.22600888949809223546343328773, 12.276477768235734591281894129758, 12.94822802294948536874200052626, 13.51520251781449915289509859476, 14.3793917205829615753718536014, 15.11486765438792745841888098824, 15.26006988859116382523833086288, 16.392530541803867306554463400600, 16.84889777083780171017513422272, 17.90526565368660156711558604240, 19.13022749193885639487640048618, 19.51878064977169690210432986297, 20.3918594080788227861095685240

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