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Properties

Label 1-1925-1925.1152-r0-0-0
Degree $1$
Conductor $1925$
Sign $0.972 + 0.234i$
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  + (−0.406 − 0.913i)2-s + (−0.866 + 0.5i)3-s + (−0.669 + 0.743i)4-s + (0.809 + 0.587i)6-s + (0.951 + 0.309i)8-s + (0.5 − 0.866i)9-s + (0.207 − 0.978i)12-s + i·13-s + (−0.104 − 0.994i)16-s + (0.994 + 0.104i)17-s + (−0.994 − 0.104i)18-s + (0.669 + 0.743i)19-s + (0.743 − 0.669i)23-s + (−0.978 + 0.207i)24-s + (0.913 − 0.406i)26-s + i·27-s + ⋯
L(s)  = 1  + (−0.406 − 0.913i)2-s + (−0.866 + 0.5i)3-s + (−0.669 + 0.743i)4-s + (0.809 + 0.587i)6-s + (0.951 + 0.309i)8-s + (0.5 − 0.866i)9-s + (0.207 − 0.978i)12-s + i·13-s + (−0.104 − 0.994i)16-s + (0.994 + 0.104i)17-s + (−0.994 − 0.104i)18-s + (0.669 + 0.743i)19-s + (0.743 − 0.669i)23-s + (−0.978 + 0.207i)24-s + (0.913 − 0.406i)26-s + i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $0.972 + 0.234i$
Analytic conductor: \(8.93966\)
Root analytic conductor: \(8.93966\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (1152, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1925,\ (0:\ ),\ 0.972 + 0.234i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9216757409 + 0.1095185462i\)
\(L(\frac12)\) \(\approx\) \(0.9216757409 + 0.1095185462i\)
\(L(1)\) \(\approx\) \(0.7009774773 - 0.08081495355i\)
\(L(1)\) \(\approx\) \(0.7009774773 - 0.08081495355i\)

Euler product

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

−19.49369553333130598934148269520, −19.24317420676683832490413511854, −18.263239717840984676003927975594, −17.664853549740205394057937393116, −17.27737769380377707687124278958, −16.40733563152446386089712597093, −15.736193970973150824669310855083, −15.168013191876860271565675015575, −14.05174853546734806715155178492, −13.50353490742416491907355519682, −12.68125148380980475538092305263, −11.895724737283545113573993984505, −10.92995480592012845313356993478, −10.27996320554450196300441525403, −9.53520800892721560159950248469, −8.52192418492316214982512009208, −7.663515872955234349154359105418, −7.20733949068727552451137907019, −6.35147789658494919900883988407, −5.36851710137947121687676638616, −5.25046640802892763939998781269, −4.0005835952615790791078863182, −2.70715242666668533339312282028, −1.30866912487161805364056401067, −0.63329741173100381092379224074, 0.88663826853394607021391479870, 1.67170800608460240748191228324, 2.98955012291323443760089290819, 3.726732146691020978783073884791, 4.60769587473262984004113147526, 5.24593994929865853164286394239, 6.33996044597440729984280081737, 7.19991351319779528858197590628, 8.20531351864719504875631868981, 9.06222370586080119282776871718, 9.897623559179765911071691528852, 10.23744953404741256754604905978, 11.299632327127669072600189124409, 11.67492319764587479891838852720, 12.42579463328577805193384295985, 13.10843421859581078366848219604, 14.20661655786862701602666525795, 14.799162654952134583466022993156, 16.21561580101791135835905095974, 16.40727005245321712363903281806, 17.20688730923124549064838006688, 17.93776746231905719722783974463, 18.686154098627534609775039812264, 19.14274168548533581265843755584, 20.20780170206033221201625474905

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