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Properties

Label 1-1925-1925.1139-r0-0-0
Degree $1$
Conductor $1925$
Sign $0.957 - 0.289i$
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  + (0.669 − 0.743i)2-s + (0.669 + 0.743i)3-s + (−0.104 − 0.994i)4-s + 6-s + (−0.809 − 0.587i)8-s + (−0.104 + 0.994i)9-s + (0.669 − 0.743i)12-s + (0.809 + 0.587i)13-s + (−0.978 + 0.207i)16-s + (0.5 − 0.866i)17-s + (0.669 + 0.743i)18-s + (−0.104 + 0.994i)19-s + (0.978 − 0.207i)23-s + (−0.104 − 0.994i)24-s + (0.978 − 0.207i)26-s + (−0.809 + 0.587i)27-s + ⋯
L(s)  = 1  + (0.669 − 0.743i)2-s + (0.669 + 0.743i)3-s + (−0.104 − 0.994i)4-s + 6-s + (−0.809 − 0.587i)8-s + (−0.104 + 0.994i)9-s + (0.669 − 0.743i)12-s + (0.809 + 0.587i)13-s + (−0.978 + 0.207i)16-s + (0.5 − 0.866i)17-s + (0.669 + 0.743i)18-s + (−0.104 + 0.994i)19-s + (0.978 − 0.207i)23-s + (−0.104 − 0.994i)24-s + (0.978 − 0.207i)26-s + (−0.809 + 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $0.957 - 0.289i$
Analytic conductor: \(8.93966\)
Root analytic conductor: \(8.93966\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (1139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1925,\ (0:\ ),\ 0.957 - 0.289i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.039577860 - 0.4489156076i\)
\(L(\frac12)\) \(\approx\) \(3.039577860 - 0.4489156076i\)
\(L(1)\) \(\approx\) \(1.858217330 - 0.3301610705i\)
\(L(1)\) \(\approx\) \(1.858217330 - 0.3301610705i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.669 - 0.743i)T \)
3 \( 1 + (0.669 + 0.743i)T \)
13 \( 1 + (0.809 + 0.587i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.104 + 0.994i)T \)
23 \( 1 + (0.978 - 0.207i)T \)
29 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (-0.669 - 0.743i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.978 + 0.207i)T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.913 + 0.406i)T \)
67 \( 1 + (0.104 + 0.994i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (-0.913 + 0.406i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 - T \)
89 \( 1 + (0.978 - 0.207i)T \)
97 \( 1 + T \)
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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

−20.04000264684223425460949202574, −19.44153533070354024350471981654, −18.45310539657539044830540881819, −17.81686673726927535965091540691, −17.23837551032669392356340255761, −16.25340618407538685080704546568, −15.47348784930291628121571978740, −14.87479899303076683247002955388, −14.211092492730865136452530557980, −13.42917895305370059541861393490, −12.89261906455174620254580550855, −12.336091664696889430778199005414, −11.36558683390990340132868704595, −10.45971461261695931960154756042, −9.01772504988459095023661887089, −8.73796341996729168323067211424, −7.815340010236385156999352329699, −7.18033184203916382491394119389, −6.42382102071360953641277063010, −5.69635620986597738754391272316, −4.76167450477185923960104492529, −3.61157194320144891026910978536, −3.16777309003694296216993004276, −2.136078979278182138378164643556, −0.924308824632893883426861570689, 1.083366485498559415499543017133, 2.113469823014319792260530973, 2.96300587664758407722027967383, 3.68046199192716915342628953327, 4.4018725846597441764136702987, 5.16983466661452132230849848776, 6.01464647011668952986136505996, 7.01544919545341284345242027555, 8.193171092059557514687579920907, 8.90040240759619397251867223503, 9.81829960465683821442801622954, 10.14282513015141186406584196489, 11.31442934225142749262618886088, 11.54794467375734631363874862741, 12.81423477862552327159305764017, 13.37102709788405875047787582016, 14.23585783805776824128682464182, 14.5783152853958527129836011557, 15.47811939853802839733667127144, 16.15200043746093104359459784548, 16.83651425312069092783633109181, 18.154166764392559897553142480314, 18.89543465598089748304813193407, 19.29519462520034282261089493378, 20.40047105950207957849962646015

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