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Properties

Label 1-35e2-1225.316-r0-0-0
Degree $1$
Conductor $1225$
Sign $0.939 + 0.341i$
Analytic cond. $5.68887$
Root an. cond. $5.68887$
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  + (0.936 + 0.351i)2-s + (0.983 − 0.178i)3-s + (0.753 + 0.657i)4-s + (0.983 + 0.178i)6-s + (0.473 + 0.880i)8-s + (0.936 − 0.351i)9-s + (0.936 + 0.351i)11-s + (0.858 + 0.512i)12-s + (−0.550 − 0.834i)13-s + (0.134 + 0.990i)16-s + (0.753 − 0.657i)17-s + 18-s + (−0.809 − 0.587i)19-s + (0.753 + 0.657i)22-s + (0.858 − 0.512i)23-s + (0.623 + 0.781i)24-s + ⋯
L(s)  = 1  + (0.936 + 0.351i)2-s + (0.983 − 0.178i)3-s + (0.753 + 0.657i)4-s + (0.983 + 0.178i)6-s + (0.473 + 0.880i)8-s + (0.936 − 0.351i)9-s + (0.936 + 0.351i)11-s + (0.858 + 0.512i)12-s + (−0.550 − 0.834i)13-s + (0.134 + 0.990i)16-s + (0.753 − 0.657i)17-s + 18-s + (−0.809 − 0.587i)19-s + (0.753 + 0.657i)22-s + (0.858 − 0.512i)23-s + (0.623 + 0.781i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $0.939 + 0.341i$
Analytic conductor: \(5.68887\)
Root analytic conductor: \(5.68887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (0:\ ),\ 0.939 + 0.341i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.276679263 + 0.7534656897i\)
\(L(\frac12)\) \(\approx\) \(4.276679263 + 0.7534656897i\)
\(L(1)\) \(\approx\) \(2.618131144 + 0.3933392038i\)
\(L(1)\) \(\approx\) \(2.618131144 + 0.3933392038i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.936 + 0.351i)T \)
3 \( 1 + (0.983 - 0.178i)T \)
11 \( 1 + (0.936 + 0.351i)T \)
13 \( 1 + (-0.550 - 0.834i)T \)
17 \( 1 + (0.753 - 0.657i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (0.858 - 0.512i)T \)
29 \( 1 + (-0.393 - 0.919i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (0.858 + 0.512i)T \)
41 \( 1 + (-0.691 + 0.722i)T \)
43 \( 1 + (-0.900 + 0.433i)T \)
47 \( 1 + (-0.0448 + 0.998i)T \)
53 \( 1 + (0.753 + 0.657i)T \)
59 \( 1 + (0.134 + 0.990i)T \)
61 \( 1 + (-0.995 + 0.0896i)T \)
67 \( 1 + (-0.809 - 0.587i)T \)
71 \( 1 + (0.753 + 0.657i)T \)
73 \( 1 + (-0.550 + 0.834i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (-0.0448 - 0.998i)T \)
89 \( 1 + (-0.550 + 0.834i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−21.34462502960485653720147779091, −20.35875931044819453581722727912, −19.61529534559161451783801339850, −19.19175140635865853326776344684, −18.45021572599663053756790149148, −16.79856886961685761469561749929, −16.5103128106320211424615867040, −15.28004540519860276853678502497, −14.65694686526450294775851435171, −14.29727891407742501554868006174, −13.40624610264399034475140431720, −12.63303772297074861471895770109, −11.92842418833760659043370890880, −10.93720520827454898745114490057, −10.1465923714502363543417532078, −9.299279804723538088712401215767, −8.54053506354174248198252732807, −7.3193853879335668386206781438, −6.73362665750672221387222250730, −5.618521099181862153677849949683, −4.6530256163277291747971407150, −3.73024954760437288931391267079, −3.325363499440872277326230389745, −2.01641876591091940012758904391, −1.47794972822337743131288894014, 1.31867907129038878391293321557, 2.51256893525376831679708747507, 3.06751067496467218641545760533, 4.12826870211166341675414593168, 4.76946696478313173490920297227, 5.93757868294990280052513136705, 6.86656438530182515205326562279, 7.50270244639758077720916112488, 8.27529927816986350032756805323, 9.23311951476009354636835697171, 10.074546148806041803990033275443, 11.2356290836113121751358196129, 12.10759879290734750917782812893, 12.86573149576179969709003346783, 13.414653245460706846310959628224, 14.33850232955734848250256713739, 15.02082793363722771235095035324, 15.2377344975830564958520835057, 16.54330116050552435843940324727, 17.074207985219454408216738014831, 18.132076812343113784592612058119, 19.1002786159763036718045590484, 19.95180183820334763526778817351, 20.362228592234767443146161716977, 21.21889592149767361538587279840

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