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Properties

Label 1-35e2-1225.34-r1-0-0
Degree $1$
Conductor $1225$
Sign $-0.902 - 0.430i$
Analytic cond. $131.644$
Root an. cond. $131.644$
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  + (−0.134 + 0.990i)2-s + (0.753 + 0.657i)3-s + (−0.963 − 0.266i)4-s + (−0.753 + 0.657i)6-s + (0.393 − 0.919i)8-s + (0.134 + 0.990i)9-s + (0.134 − 0.990i)11-s + (−0.550 − 0.834i)12-s + (−0.691 + 0.722i)13-s + (0.858 + 0.512i)16-s + (−0.963 + 0.266i)17-s − 18-s + (0.809 + 0.587i)19-s + (0.963 + 0.266i)22-s + (0.550 − 0.834i)23-s + (0.900 − 0.433i)24-s + ⋯
L(s)  = 1  + (−0.134 + 0.990i)2-s + (0.753 + 0.657i)3-s + (−0.963 − 0.266i)4-s + (−0.753 + 0.657i)6-s + (0.393 − 0.919i)8-s + (0.134 + 0.990i)9-s + (0.134 − 0.990i)11-s + (−0.550 − 0.834i)12-s + (−0.691 + 0.722i)13-s + (0.858 + 0.512i)16-s + (−0.963 + 0.266i)17-s − 18-s + (0.809 + 0.587i)19-s + (0.963 + 0.266i)22-s + (0.550 − 0.834i)23-s + (0.900 − 0.433i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.902 - 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.902 - 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $-0.902 - 0.430i$
Analytic conductor: \(131.644\)
Root analytic conductor: \(131.644\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (1:\ ),\ -0.902 - 0.430i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3703666952 + 1.637191636i\)
\(L(\frac12)\) \(\approx\) \(-0.3703666952 + 1.637191636i\)
\(L(1)\) \(\approx\) \(0.7700610925 + 0.7896284731i\)
\(L(1)\) \(\approx\) \(0.7700610925 + 0.7896284731i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.134 + 0.990i)T \)
3 \( 1 + (0.753 + 0.657i)T \)
11 \( 1 + (0.134 - 0.990i)T \)
13 \( 1 + (-0.691 + 0.722i)T \)
17 \( 1 + (-0.963 + 0.266i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
23 \( 1 + (0.550 - 0.834i)T \)
29 \( 1 + (-0.0448 + 0.998i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (0.550 + 0.834i)T \)
41 \( 1 + (0.995 + 0.0896i)T \)
43 \( 1 + (0.222 - 0.974i)T \)
47 \( 1 + (0.983 - 0.178i)T \)
53 \( 1 + (0.963 + 0.266i)T \)
59 \( 1 + (-0.858 - 0.512i)T \)
61 \( 1 + (-0.936 - 0.351i)T \)
67 \( 1 + (0.809 + 0.587i)T \)
71 \( 1 + (-0.963 - 0.266i)T \)
73 \( 1 + (-0.691 - 0.722i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (0.983 + 0.178i)T \)
89 \( 1 + (0.691 + 0.722i)T \)
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

−20.32022560424680821705755632748, −19.734748077778960803444002013034, −19.29820161222720431945821541918, −18.209051282200493818031379572892, −17.723922890972433181154861089489, −17.17792732741944301913182818278, −15.594034394317498339568440630133, −14.97413416450808029347116917658, −14.061987542976840471211759030735, −13.29912786009371280320650346126, −12.79111866817323064913150561786, −11.947479306372826599386318348471, −11.29029988768071490596396197703, −10.09878256626449057089065207990, −9.44014462395549230748900529178, −8.87659191455345287841706446205, −7.60283710080322298418763995168, −7.418812980579240532388925072064, −5.976409036891029743831631439185, −4.76402641749504983073695079375, −4.020949669835713413276606682556, −2.78059161725132302163579052668, −2.427782567140726913608000575354, −1.287842074674851823865514910731, −0.34639955397382518177398696498, 1.127480303965094548606266720362, 2.55208522861625171062348284355, 3.590878256430301413228524251705, 4.450769507411375847058658125184, 5.15461392800126179952936665346, 6.206912520737661596390089415599, 7.09539200051330571018540635434, 7.94953695695356428404138151787, 8.83452439358227978814283014361, 9.15496139205569667132089497834, 10.20465835727052941905968023679, 10.889368688098633873052205806635, 12.15119348043645411439279105215, 13.25933368735224730054005141757, 13.94813137220211454718673834304, 14.45807135793237645756744396463, 15.24152910072256808402237989406, 16.027037999497795032151236654419, 16.57627722644340318711855769027, 17.25485441053163226241518286866, 18.41078401108286591210145303575, 19.01072778170800340476525537191, 19.72758019914165602239577143177, 20.58545337996199673501851042687, 21.731308209356669464030813575

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