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Properties

Label 1-1925-1925.303-r0-0-0
Degree $1$
Conductor $1925$
Sign $-0.386 - 0.922i$
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  + (0.866 − 0.5i)2-s + (0.994 − 0.104i)3-s + (0.5 − 0.866i)4-s + (0.809 − 0.587i)6-s − i·8-s + (0.978 − 0.207i)9-s + (0.406 − 0.913i)12-s + (−0.951 − 0.309i)13-s + (−0.5 − 0.866i)16-s + (0.994 − 0.104i)17-s + (0.743 − 0.669i)18-s + (−0.5 − 0.866i)19-s + (−0.994 − 0.104i)23-s + (−0.104 − 0.994i)24-s + (−0.978 + 0.207i)26-s + (0.951 − 0.309i)27-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.994 − 0.104i)3-s + (0.5 − 0.866i)4-s + (0.809 − 0.587i)6-s − i·8-s + (0.978 − 0.207i)9-s + (0.406 − 0.913i)12-s + (−0.951 − 0.309i)13-s + (−0.5 − 0.866i)16-s + (0.994 − 0.104i)17-s + (0.743 − 0.669i)18-s + (−0.5 − 0.866i)19-s + (−0.994 − 0.104i)23-s + (−0.104 − 0.994i)24-s + (−0.978 + 0.207i)26-s + (0.951 − 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $-0.386 - 0.922i$
Analytic conductor: \(8.93966\)
Root analytic conductor: \(8.93966\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1925,\ (0:\ ),\ -0.386 - 0.922i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.113302926 - 3.175313940i\)
\(L(\frac12)\) \(\approx\) \(2.113302926 - 3.175313940i\)
\(L(1)\) \(\approx\) \(1.980813099 - 1.189065224i\)
\(L(1)\) \(\approx\) \(1.980813099 - 1.189065224i\)

Euler product

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

−20.498080735370223978822545487105, −19.59946071107772431973369801636, −19.05920647104695386337102543194, −18.01904364056201138967786281882, −17.15242011758578269091205544751, −16.305611934676452271635827291582, −15.83725353004856081565869710737, −14.85187319167129094706363937120, −14.35162431871867434365407231468, −14.003666837247997714522725854875, −12.888707532098890140912251961357, −12.43322525320362765780284353294, −11.6768729735465503679736574768, −10.440110676323266297321157475477, −9.80403259598101359323695353002, −8.77561304521173385401178501911, −8.010748984956787858458777238471, −7.480167643272952765787591174488, −6.63068229236851774602883268958, −5.6935791585735925680923982794, −4.77882551344351267899560960383, −4.01449293376661680892510284491, −3.30518270708054611927279335861, −2.42900746279573460284890798683, −1.63042131953126801412245641724, 0.7985888477466818502317210562, 2.01973848324662910735600721072, 2.56519606279200296005030673656, 3.442411077227497856762161299732, 4.193969110936550803258411847061, 5.02845487574828139763068342529, 5.91405828594005153320400509688, 6.99979426292182529709476808215, 7.52479698029807158848292611934, 8.583175877284796382521414013197, 9.47841158950830939773233887420, 10.13885716915753564204231092086, 10.79599425525761460767254666240, 12.05771626203213237659402702601, 12.3782505124920652875145663565, 13.242400442378202054074477654689, 13.95657128099786225575405130745, 14.51464725768895319277984401923, 15.15793801679648119589540484232, 15.82500334701423062320402968974, 16.69955869080645755988865433708, 17.86948029961029873147166801777, 18.61959910611409341666383303736, 19.46015961772563078713919382238, 19.818165170249102047060750202834

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