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Properties

Label 1-35e2-1225.141-r0-0-0
Degree $1$
Conductor $1225$
Sign $-0.959 - 0.280i$
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  + (0.858 − 0.512i)2-s + (−0.963 − 0.266i)3-s + (0.473 − 0.880i)4-s + (−0.963 + 0.266i)6-s + (−0.0448 − 0.998i)8-s + (0.858 + 0.512i)9-s + (0.858 − 0.512i)11-s + (−0.691 + 0.722i)12-s + (−0.995 − 0.0896i)13-s + (−0.550 − 0.834i)16-s + (0.473 + 0.880i)17-s + 18-s + (−0.809 − 0.587i)19-s + (0.473 − 0.880i)22-s + (−0.691 − 0.722i)23-s + (−0.222 + 0.974i)24-s + ⋯
L(s)  = 1  + (0.858 − 0.512i)2-s + (−0.963 − 0.266i)3-s + (0.473 − 0.880i)4-s + (−0.963 + 0.266i)6-s + (−0.0448 − 0.998i)8-s + (0.858 + 0.512i)9-s + (0.858 − 0.512i)11-s + (−0.691 + 0.722i)12-s + (−0.995 − 0.0896i)13-s + (−0.550 − 0.834i)16-s + (0.473 + 0.880i)17-s + 18-s + (−0.809 − 0.587i)19-s + (0.473 − 0.880i)22-s + (−0.691 − 0.722i)23-s + (−0.222 + 0.974i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.280i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.280i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $-0.959 - 0.280i$
Analytic conductor: \(5.68887\)
Root analytic conductor: \(5.68887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (0:\ ),\ -0.959 - 0.280i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1986346582 - 1.386122321i\)
\(L(\frac12)\) \(\approx\) \(0.1986346582 - 1.386122321i\)
\(L(1)\) \(\approx\) \(0.9609399071 - 0.6987647878i\)
\(L(1)\) \(\approx\) \(0.9609399071 - 0.6987647878i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.858 - 0.512i)T \)
3 \( 1 + (-0.963 - 0.266i)T \)
11 \( 1 + (0.858 - 0.512i)T \)
13 \( 1 + (-0.995 - 0.0896i)T \)
17 \( 1 + (0.473 + 0.880i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (-0.691 - 0.722i)T \)
29 \( 1 + (0.983 - 0.178i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (-0.691 + 0.722i)T \)
41 \( 1 + (0.936 - 0.351i)T \)
43 \( 1 + (0.623 - 0.781i)T \)
47 \( 1 + (0.753 + 0.657i)T \)
53 \( 1 + (0.473 - 0.880i)T \)
59 \( 1 + (-0.550 - 0.834i)T \)
61 \( 1 + (0.134 - 0.990i)T \)
67 \( 1 + (-0.809 - 0.587i)T \)
71 \( 1 + (0.473 - 0.880i)T \)
73 \( 1 + (-0.995 + 0.0896i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (0.753 - 0.657i)T \)
89 \( 1 + (-0.995 + 0.0896i)T \)
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

−21.68594048756949729893239248685, −21.089694067135533136519526267746, −20.1124124970561133313030472105, −19.296495709552340660988641342880, −17.95855288480062408766490280280, −17.53620860610367941775846034248, −16.65028387933438714791593773192, −16.22896312411807486782772799159, −15.2963049719206882619794163806, −14.56531656060384256670184883863, −13.924360394874791942343836185354, −12.68201337445214712462336692596, −12.19137866712660792170402999969, −11.676733851864111569108968485838, −10.65476049112089694358198342850, −9.77645958713496894785034114489, −8.84059488990629700981921270517, −7.43493374797093554371546975633, −7.080793859770505533836137178391, −6.04320966344636522469456611952, −5.42534097700989165839590134245, −4.46637205138608036755384418951, −3.96845247868666041680250569536, −2.70404702068995988435710487759, −1.47508361581740804953726472266, 0.46571627531042725996697850344, 1.62523373663179642858581728773, 2.51944949774650663727880772968, 3.836573639694326470130131040152, 4.500783041538273077945754702688, 5.41520579810801992692932778341, 6.22435169435518290733385895733, 6.75938606739362259322127086872, 7.83482793195075736420997926023, 9.14482370391072753671033848177, 10.182394514415368164649555776469, 10.724032456157509435408882879762, 11.55634374896693205623708276953, 12.331333887431418899584718907866, 12.66974229930381583828251707509, 13.744056303410717934141006137952, 14.46409941405600873293371167187, 15.27583032844284501440325850921, 16.174910515765991800332075653250, 16.985389899452378649237016363482, 17.582425918303284460406788016731, 18.80096692627937178865646733256, 19.2485450945520353393367406597, 19.96941095759625267102457715466, 21.05128418160615810201582235525

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