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Properties

Label 1-65e2-4225.744-r0-0-0
Degree $1$
Conductor $4225$
Sign $0.255 - 0.966i$
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  + (0.581 − 0.813i)2-s + (0.932 − 0.362i)3-s + (−0.324 − 0.945i)4-s + (0.247 − 0.968i)6-s + (0.996 + 0.0804i)7-s + (−0.958 − 0.285i)8-s + (0.737 − 0.675i)9-s + (0.737 + 0.675i)11-s + (−0.644 − 0.764i)12-s + (0.644 − 0.764i)14-s + (−0.789 + 0.613i)16-s + (0.231 + 0.972i)17-s + (−0.120 − 0.992i)18-s + (0.669 − 0.743i)19-s + (0.958 − 0.285i)21-s + (0.978 − 0.207i)22-s + ⋯
L(s)  = 1  + (0.581 − 0.813i)2-s + (0.932 − 0.362i)3-s + (−0.324 − 0.945i)4-s + (0.247 − 0.968i)6-s + (0.996 + 0.0804i)7-s + (−0.958 − 0.285i)8-s + (0.737 − 0.675i)9-s + (0.737 + 0.675i)11-s + (−0.644 − 0.764i)12-s + (0.644 − 0.764i)14-s + (−0.789 + 0.613i)16-s + (0.231 + 0.972i)17-s + (−0.120 − 0.992i)18-s + (0.669 − 0.743i)19-s + (0.958 − 0.285i)21-s + (0.978 − 0.207i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.255 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.255 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4225\)    =    \(5^{2} \cdot 13^{2}\)
Sign: $0.255 - 0.966i$
Analytic conductor: \(19.6208\)
Root analytic conductor: \(19.6208\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4225} (744, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4225,\ (0:\ ),\ 0.255 - 0.966i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.458599269 - 2.662569691i\)
\(L(\frac12)\) \(\approx\) \(3.458599269 - 2.662569691i\)
\(L(1)\) \(\approx\) \(1.946723942 - 1.144108754i\)
\(L(1)\) \(\approx\) \(1.946723942 - 1.144108754i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.581 - 0.813i)T \)
3 \( 1 + (0.932 - 0.362i)T \)
7 \( 1 + (0.996 + 0.0804i)T \)
11 \( 1 + (0.737 + 0.675i)T \)
17 \( 1 + (0.231 + 0.972i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 + (0.104 + 0.994i)T \)
29 \( 1 + (-0.00805 - 0.999i)T \)
31 \( 1 + (0.715 + 0.698i)T \)
37 \( 1 + (-0.619 + 0.784i)T \)
41 \( 1 + (0.0563 + 0.998i)T \)
43 \( 1 + (0.0402 + 0.999i)T \)
47 \( 1 + (0.906 + 0.421i)T \)
53 \( 1 + (0.607 + 0.794i)T \)
59 \( 1 + (0.339 - 0.940i)T \)
61 \( 1 + (0.184 + 0.982i)T \)
67 \( 1 + (0.324 - 0.945i)T \)
71 \( 1 + (0.152 - 0.988i)T \)
73 \( 1 + (-0.215 + 0.976i)T \)
79 \( 1 + (-0.906 - 0.421i)T \)
83 \( 1 + (-0.779 + 0.626i)T \)
89 \( 1 + (-0.104 - 0.994i)T \)
97 \( 1 + (-0.899 + 0.435i)T \)
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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

−18.56045029550665741138869906955, −17.734729722740055008079065204922, −16.93830079815956899194510183615, −16.31153636137633207039654951498, −15.80263141885095133969132187217, −14.92667908105677097507078166976, −14.42556703882452952926185549963, −13.9607489472765631378471358257, −13.55013703562763453632635003338, −12.44114059412076829139099042222, −11.88669823544198375863261938444, −11.05429315315676521956494755165, −10.17712995993014670673525077185, −9.23042743902202750850051950150, −8.63279908407411851651456381503, −8.200522656574495487460818386093, −7.28457369122293354160563975462, −6.91842080813435832111594931214, −5.61064265129306483574481575849, −5.21251517000494381760614730953, −4.211006831199472678093651544264, −3.816117442890483065501359593918, −2.919369061206407163593122242886, −2.14359611014564348029453218388, −0.962531190205426847446253250617, 1.31232120511787409347278009333, 1.377798091499802117552695537813, 2.42821338337893252681770504201, 3.09200150653311087735185513308, 4.03160631988060315947760045798, 4.48350998184173399289316354199, 5.36723416219835977352351144394, 6.30419907323949572657821870793, 7.07817156918170162680795681075, 7.93534017354426033625947161961, 8.60809884165634570109375566411, 9.415864268061661053438224184259, 9.88497814439650718822976913199, 10.78959244227417718028183546340, 11.65005660786737807156919929212, 12.04784455354551244438362515023, 12.81564304854584564967122375805, 13.609590472185067616961621764325, 13.99067236241226735883195166023, 14.79856574216948674965224203622, 15.13900686918259088909890104731, 15.79497847404499793000478287990, 17.328219883224011379960216233569, 17.60859446767734324676615891843, 18.440961274687725264776257690614

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