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Properties

Label 1-35e2-1225.806-r0-0-0
Degree $1$
Conductor $1225$
Sign $-0.0909 - 0.995i$
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  + (−0.0448 + 0.998i)2-s + (−0.691 + 0.722i)3-s + (−0.995 − 0.0896i)4-s + (−0.691 − 0.722i)6-s + (0.134 − 0.990i)8-s + (−0.0448 − 0.998i)9-s + (−0.0448 + 0.998i)11-s + (0.753 − 0.657i)12-s + (−0.963 + 0.266i)13-s + (0.983 + 0.178i)16-s + (−0.995 + 0.0896i)17-s + 18-s + (0.309 + 0.951i)19-s + (−0.995 − 0.0896i)22-s + (0.753 + 0.657i)23-s + (0.623 + 0.781i)24-s + ⋯
L(s)  = 1  + (−0.0448 + 0.998i)2-s + (−0.691 + 0.722i)3-s + (−0.995 − 0.0896i)4-s + (−0.691 − 0.722i)6-s + (0.134 − 0.990i)8-s + (−0.0448 − 0.998i)9-s + (−0.0448 + 0.998i)11-s + (0.753 − 0.657i)12-s + (−0.963 + 0.266i)13-s + (0.983 + 0.178i)16-s + (−0.995 + 0.0896i)17-s + 18-s + (0.309 + 0.951i)19-s + (−0.995 − 0.0896i)22-s + (0.753 + 0.657i)23-s + (0.623 + 0.781i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0909 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0909 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $-0.0909 - 0.995i$
Analytic conductor: \(5.68887\)
Root analytic conductor: \(5.68887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (806, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (0:\ ),\ -0.0909 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2654850569 + 0.2908264390i\)
\(L(\frac12)\) \(\approx\) \(-0.2654850569 + 0.2908264390i\)
\(L(1)\) \(\approx\) \(0.3936035973 + 0.4809801132i\)
\(L(1)\) \(\approx\) \(0.3936035973 + 0.4809801132i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.0448 + 0.998i)T \)
3 \( 1 + (-0.691 + 0.722i)T \)
11 \( 1 + (-0.0448 + 0.998i)T \)
13 \( 1 + (-0.963 + 0.266i)T \)
17 \( 1 + (-0.995 + 0.0896i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (0.753 + 0.657i)T \)
29 \( 1 + (0.858 + 0.512i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (0.753 - 0.657i)T \)
41 \( 1 + (0.473 + 0.880i)T \)
43 \( 1 + (-0.900 + 0.433i)T \)
47 \( 1 + (-0.550 - 0.834i)T \)
53 \( 1 + (-0.995 - 0.0896i)T \)
59 \( 1 + (0.983 + 0.178i)T \)
61 \( 1 + (-0.393 - 0.919i)T \)
67 \( 1 + (0.309 + 0.951i)T \)
71 \( 1 + (-0.995 - 0.0896i)T \)
73 \( 1 + (-0.963 - 0.266i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (-0.550 + 0.834i)T \)
89 \( 1 + (-0.963 - 0.266i)T \)
97 \( 1 + (0.309 - 0.951i)T \)
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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

−20.46649894009204306918153318172, −19.60890357623088852084244256608, −19.133846297667183130708000753034, −18.40300477840248686366357347202, −17.55477603114586707518430779662, −17.14509443265064093063469368427, −16.179971732459180531835027537862, −15.04684748858459084565691409025, −13.959966891153838328281284375073, −13.34250865814446377731476577811, −12.75779883627344465230048970729, −11.826911338485045883893064593205, −11.2845879499910837387436390090, −10.63488693717311229253647107027, −9.69038891026641552298701504084, −8.708591477096377045985246982776, −7.95670158605359542509878268750, −6.935520890960137208286435382473, −5.99016582362443436122430989474, −5.01085438191919371507200419545, −4.39821172581524656216423148492, −2.88335958801959324336904701860, −2.4037340339645271722193083499, −1.08327326631538036592713743176, −0.21038221879617620955082432260, 1.43693064678097469464196150585, 3.0909048685347810423168777507, 4.276449707195092102358824567274, 4.7882677893554975743912064157, 5.52588460444304377944263891001, 6.59547790228238199675146140518, 7.08949999328998728462015733772, 8.165609170043915522344438881365, 9.19999437071883206957577551178, 9.78124828308866885455190802321, 10.45548169024419728833245733954, 11.60012037293232785749241604654, 12.42996077195296907071869289566, 13.153844164111279714212783761556, 14.44301749201678658568837447750, 14.80376962645462322181814735591, 15.66518891713806220839597341521, 16.27013847104442137610550945129, 17.04120246186008496382450300842, 17.72515868143756204987999891399, 18.136349737373426818120275557523, 19.36735311138256512004013480280, 20.13004396985228126751377502613, 21.26469615950868327070364143042, 21.85038065857594875499916141396

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