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Properties

Label 1-65e2-4225.1633-r0-0-0
Degree $1$
Conductor $4225$
Sign $-0.361 - 0.932i$
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  + (0.644 − 0.764i)2-s + (−0.192 + 0.981i)3-s + (−0.168 − 0.985i)4-s + (0.626 + 0.779i)6-s + (0.748 − 0.663i)7-s + (−0.861 − 0.506i)8-s + (−0.926 − 0.377i)9-s + (−0.377 − 0.926i)11-s + (0.999 + 0.0241i)12-s + (−0.0241 − 0.999i)14-s + (−0.943 + 0.331i)16-s + (0.506 − 0.861i)17-s + (−0.885 + 0.464i)18-s + (0.951 − 0.309i)19-s + (0.506 + 0.861i)21-s + (−0.951 − 0.309i)22-s + ⋯
L(s)  = 1  + (0.644 − 0.764i)2-s + (−0.192 + 0.981i)3-s + (−0.168 − 0.985i)4-s + (0.626 + 0.779i)6-s + (0.748 − 0.663i)7-s + (−0.861 − 0.506i)8-s + (−0.926 − 0.377i)9-s + (−0.377 − 0.926i)11-s + (0.999 + 0.0241i)12-s + (−0.0241 − 0.999i)14-s + (−0.943 + 0.331i)16-s + (0.506 − 0.861i)17-s + (−0.885 + 0.464i)18-s + (0.951 − 0.309i)19-s + (0.506 + 0.861i)21-s + (−0.951 − 0.309i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.361 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.361 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4225\)    =    \(5^{2} \cdot 13^{2}\)
Sign: $-0.361 - 0.932i$
Analytic conductor: \(19.6208\)
Root analytic conductor: \(19.6208\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4225} (1633, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4225,\ (0:\ ),\ -0.361 - 0.932i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.328649692 - 1.941045905i\)
\(L(\frac12)\) \(\approx\) \(1.328649692 - 1.941045905i\)
\(L(1)\) \(\approx\) \(1.300811816 - 0.6269412870i\)
\(L(1)\) \(\approx\) \(1.300811816 - 0.6269412870i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.644 - 0.764i)T \)
3 \( 1 + (-0.192 + 0.981i)T \)
7 \( 1 + (0.748 - 0.663i)T \)
11 \( 1 + (-0.377 - 0.926i)T \)
17 \( 1 + (0.506 - 0.861i)T \)
19 \( 1 + (0.951 - 0.309i)T \)
23 \( 1 + (0.587 + 0.809i)T \)
29 \( 1 + (0.0724 - 0.997i)T \)
31 \( 1 + (0.626 + 0.779i)T \)
37 \( 1 + (0.262 - 0.964i)T \)
41 \( 1 + (0.873 + 0.485i)T \)
43 \( 1 + (0.935 + 0.354i)T \)
47 \( 1 + (-0.715 + 0.698i)T \)
53 \( 1 + (-0.916 + 0.399i)T \)
59 \( 1 + (0.999 + 0.0241i)T \)
61 \( 1 + (0.995 + 0.0965i)T \)
67 \( 1 + (-0.168 + 0.985i)T \)
71 \( 1 + (-0.192 + 0.981i)T \)
73 \( 1 + (0.926 - 0.377i)T \)
79 \( 1 + (-0.715 + 0.698i)T \)
83 \( 1 + (-0.981 + 0.192i)T \)
89 \( 1 + (-0.587 - 0.809i)T \)
97 \( 1 + (-0.607 - 0.794i)T \)
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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

−18.43835654595560951820578733180, −17.77947636640793931925930183345, −17.329378307910201776955253429041, −16.61061169545044645150096834553, −15.80969042939182142733123565070, −14.99500959556285598626464048149, −14.542671211561288237578612207218, −13.96754281330112468920540319659, −12.99875763759353820085201733319, −12.630721972619518469302473117967, −11.99016033923535154944231779097, −11.43413748090685082105718080408, −10.50400404656701145945725591261, −9.379724536505791098300311391463, −8.530816247759199295373151180039, −7.96074645525885942379244737997, −7.46787271077485121776884229968, −6.65738443581480548458701236419, −6.0025139903810188845318286449, −5.20991707468607431816249345426, −4.83508523950680745501751680099, −3.69910981625012628127225802035, −2.71188901480059364432551963321, −2.10988619125648688883463289626, −1.10328270421544349525333529717, 0.60868259852557367543328248671, 1.27895903909207856896498894708, 2.73812757397216694631686880184, 3.07368152857678827115956596449, 4.050501098790614637739138907062, 4.55834686776497285451217448056, 5.459989952262464832803476738, 5.6612593459992794926524681886, 6.87673303407528615236061893527, 7.814261446056436196486118571529, 8.68016263995843880159482038497, 9.5753237263708274389941572154, 9.94191721519256239648310262553, 10.87015790074391973510651422034, 11.354356897383759403638068784682, 11.61542563225341582765719378373, 12.72754690873908181539626221355, 13.54888849752856779735060151673, 14.26402561217474640085375444643, 14.37753690199071327097405300953, 15.60466556958315500581862939857, 15.87476596788433318061743473726, 16.69509876334526029147148216846, 17.67017391588102925763628587690, 18.01720741002584476305842519521

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