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Properties

Label 1-35e2-1225.1219-r0-0-0
Degree $1$
Conductor $1225$
Sign $0.154 + 0.987i$
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  + (−0.753 − 0.657i)2-s + (−0.936 + 0.351i)3-s + (0.134 + 0.990i)4-s + (0.936 + 0.351i)6-s + (0.550 − 0.834i)8-s + (0.753 − 0.657i)9-s + (0.753 + 0.657i)11-s + (−0.473 − 0.880i)12-s + (0.393 − 0.919i)13-s + (−0.963 + 0.266i)16-s + (−0.134 + 0.990i)17-s − 18-s + (0.309 + 0.951i)19-s + (−0.134 − 0.990i)22-s + (−0.473 + 0.880i)23-s + (−0.222 + 0.974i)24-s + ⋯
L(s)  = 1  + (−0.753 − 0.657i)2-s + (−0.936 + 0.351i)3-s + (0.134 + 0.990i)4-s + (0.936 + 0.351i)6-s + (0.550 − 0.834i)8-s + (0.753 − 0.657i)9-s + (0.753 + 0.657i)11-s + (−0.473 − 0.880i)12-s + (0.393 − 0.919i)13-s + (−0.963 + 0.266i)16-s + (−0.134 + 0.990i)17-s − 18-s + (0.309 + 0.951i)19-s + (−0.134 − 0.990i)22-s + (−0.473 + 0.880i)23-s + (−0.222 + 0.974i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.154 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.154 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $0.154 + 0.987i$
Analytic conductor: \(5.68887\)
Root analytic conductor: \(5.68887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (1219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (0:\ ),\ 0.154 + 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4410853554 + 0.3774567153i\)
\(L(\frac12)\) \(\approx\) \(0.4410853554 + 0.3774567153i\)
\(L(1)\) \(\approx\) \(0.5696144203 + 0.02477809719i\)
\(L(1)\) \(\approx\) \(0.5696144203 + 0.02477809719i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.753 - 0.657i)T \)
3 \( 1 + (-0.936 + 0.351i)T \)
11 \( 1 + (0.753 + 0.657i)T \)
13 \( 1 + (0.393 - 0.919i)T \)
17 \( 1 + (-0.134 + 0.990i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (-0.473 + 0.880i)T \)
29 \( 1 + (-0.691 + 0.722i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (-0.473 - 0.880i)T \)
41 \( 1 + (-0.0448 - 0.998i)T \)
43 \( 1 + (-0.623 + 0.781i)T \)
47 \( 1 + (0.995 + 0.0896i)T \)
53 \( 1 + (-0.134 - 0.990i)T \)
59 \( 1 + (-0.963 + 0.266i)T \)
61 \( 1 + (0.983 - 0.178i)T \)
67 \( 1 + (-0.309 - 0.951i)T \)
71 \( 1 + (0.134 + 0.990i)T \)
73 \( 1 + (0.393 + 0.919i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (0.995 - 0.0896i)T \)
89 \( 1 + (-0.393 - 0.919i)T \)
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

−20.8548738121696490493221364916, −19.98651541095894312370862340628, −18.98004457458204356278757714912, −18.64243642825275974424166821722, −17.8715670603170778868040370775, −16.95458998216251881255889945854, −16.61967198467711496464093639794, −15.84418822676861677380108760425, −15.04899498441700452647488001243, −13.76354727040604969924226555387, −13.609504096999703652798181343928, −12.03887202511478624935095656603, −11.453572400090790697291943254362, −10.8737623941145178554005775575, −9.78601510785845913344284492187, −9.12323548074604815592420243005, −8.191588924796159021593894709819, −7.21354350417572930563964190375, −6.56200013732367198334847313284, −5.98218887495784071015926910615, −4.981660735581688525725930660362, −4.17098091366726549477270044124, −2.45153397415797343145203605835, −1.365168586710645685605498960, −0.40674359945746369566944972529, 1.15177047148498046520089853932, 1.866354763556961919736724284354, 3.517853857699969810097633927472, 3.888247380243108084857809376102, 5.14173256484549996520577881941, 6.08261284746304007268998153987, 7.00797630962947857539412328767, 7.85668030215350539328564610297, 8.867682738417877308189193654317, 9.69718594601392747358176685191, 10.404468468123448534598898066554, 10.94434181090255543802578412416, 11.928670603850049859166636030208, 12.41233907599367062017177877907, 13.1369940924641794570679552855, 14.43513787694196807433765407322, 15.46650636520861054189451792385, 16.09473110717309242860063017325, 16.97172060571265780620725568497, 17.563393657335210752122548787857, 18.046554489930033238439728211, 18.94824322948131427152642560441, 19.840744067259868896917551904186, 20.47544654107160213024170399989, 21.30035988903442065917872791903

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