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Properties

Label 1-65e2-4225.613-r0-0-0
Degree $1$
Conductor $4225$
Sign $-0.241 + 0.970i$
Analytic cond. $19.6208$
Root an. cond. $19.6208$
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  + (−0.152 − 0.988i)2-s + (0.478 − 0.877i)3-s + (−0.953 + 0.301i)4-s + (−0.940 − 0.339i)6-s + (−0.0402 − 0.999i)7-s + (0.443 + 0.896i)8-s + (−0.541 − 0.840i)9-s + (0.840 + 0.541i)11-s + (−0.192 + 0.981i)12-s + (−0.981 + 0.192i)14-s + (0.818 − 0.574i)16-s + (−0.832 − 0.554i)17-s + (−0.748 + 0.663i)18-s + (−0.994 − 0.104i)19-s + (−0.896 − 0.443i)21-s + (0.406 − 0.913i)22-s + ⋯
L(s)  = 1  + (−0.152 − 0.988i)2-s + (0.478 − 0.877i)3-s + (−0.953 + 0.301i)4-s + (−0.940 − 0.339i)6-s + (−0.0402 − 0.999i)7-s + (0.443 + 0.896i)8-s + (−0.541 − 0.840i)9-s + (0.840 + 0.541i)11-s + (−0.192 + 0.981i)12-s + (−0.981 + 0.192i)14-s + (0.818 − 0.574i)16-s + (−0.832 − 0.554i)17-s + (−0.748 + 0.663i)18-s + (−0.994 − 0.104i)19-s + (−0.896 − 0.443i)21-s + (0.406 − 0.913i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.241 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.241 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4225\)    =    \(5^{2} \cdot 13^{2}\)
Sign: $-0.241 + 0.970i$
Analytic conductor: \(19.6208\)
Root analytic conductor: \(19.6208\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4225} (613, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4225,\ (0:\ ),\ -0.241 + 0.970i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.6367100478 - 0.8143621322i\)
\(L(\frac12)\) \(\approx\) \(-0.6367100478 - 0.8143621322i\)
\(L(1)\) \(\approx\) \(0.5213676005 - 0.7846843868i\)
\(L(1)\) \(\approx\) \(0.5213676005 - 0.7846843868i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.152 - 0.988i)T \)
3 \( 1 + (0.478 - 0.877i)T \)
7 \( 1 + (-0.0402 - 0.999i)T \)
11 \( 1 + (0.840 + 0.541i)T \)
17 \( 1 + (-0.832 - 0.554i)T \)
19 \( 1 + (-0.994 - 0.104i)T \)
23 \( 1 + (-0.207 + 0.978i)T \)
29 \( 1 + (0.892 - 0.450i)T \)
31 \( 1 + (0.764 - 0.644i)T \)
37 \( 1 + (-0.471 - 0.881i)T \)
41 \( 1 + (-0.128 - 0.991i)T \)
43 \( 1 + (0.721 - 0.692i)T \)
47 \( 1 + (0.995 + 0.0965i)T \)
53 \( 1 + (-0.144 - 0.989i)T \)
59 \( 1 + (-0.945 + 0.324i)T \)
61 \( 1 + (0.247 - 0.968i)T \)
67 \( 1 + (0.953 + 0.301i)T \)
71 \( 1 + (-0.520 - 0.853i)T \)
73 \( 1 + (0.998 + 0.0483i)T \)
79 \( 1 + (-0.995 - 0.0965i)T \)
83 \( 1 + (0.0241 - 0.999i)T \)
89 \( 1 + (-0.207 + 0.978i)T \)
97 \( 1 + (-0.513 + 0.857i)T \)
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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

−18.8515231994590066099111590157, −18.11473761405704748583606466515, −17.19128847240320650801463447512, −16.83048132344912767788381500036, −15.944484900011885085585223946615, −15.547248608097858108503394738453, −14.91209255578298064515502887655, −14.32971216564618534761848267488, −13.802859027694654407080904832428, −12.85880667327432743843715720352, −12.171646947547510621359158112300, −11.119651554517657208732657835958, −10.44806453839749824529783789752, −9.704015057083749213681635090198, −8.92204077466642999572606026492, −8.52458820830395531711833413278, −8.16239150573618365505239718070, −6.81218859270408982749465995973, −6.25866184411038583730540395435, −5.64047398847375048469044395203, −4.55463574209791356631124066905, −4.35814205104525040645374658190, −3.25947559054377497758753837177, −2.47452938542794575754887132607, −1.32091057382313445166688038005, 0.30600725306467287507177665667, 1.11419907629833344931255485349, 2.00079968687447133445219125116, 2.50258311850474299289433027524, 3.66525462800882824128663282673, 4.0327207952462510309212207250, 4.8977939132391614065132643553, 6.110467221842767748840428637063, 6.898915675419388439421337508634, 7.49908391481402547414553516968, 8.261629213244513163575240752761, 9.03934893688373008289998177006, 9.54212808021086951335363686610, 10.39803909943050659157948610923, 11.12249197532173780491602447754, 11.82660217605614772440398867257, 12.41299512551290790368553716809, 13.09980517178314637109526854707, 13.854485293688068826138490279078, 14.00418336197935993441015602215, 14.954370277833910275214516595583, 15.82871687176439847066797277534, 17.13601726690801076052175846122, 17.36447469797650380046383923261, 17.82495710077863480534205991007

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