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Properties

Label 1-4235-4235.1118-r1-0-0
Degree $1$
Conductor $4235$
Sign $0.861 + 0.508i$
Analytic cond. $455.113$
Root an. cond. $455.113$
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  + (−0.901 − 0.432i)2-s + (−0.406 − 0.913i)3-s + (0.625 + 0.780i)4-s + (−0.0285 + 0.999i)6-s + (−0.226 − 0.974i)8-s + (−0.669 + 0.743i)9-s + (0.458 − 0.888i)12-s + (−0.633 + 0.774i)13-s + (−0.217 + 0.976i)16-s + (0.556 + 0.830i)17-s + (0.924 − 0.380i)18-s + (−0.999 + 0.0380i)19-s + (−0.0950 − 0.995i)23-s + (−0.797 + 0.603i)24-s + (0.905 − 0.424i)26-s + (0.951 + 0.309i)27-s + ⋯
L(s)  = 1  + (−0.901 − 0.432i)2-s + (−0.406 − 0.913i)3-s + (0.625 + 0.780i)4-s + (−0.0285 + 0.999i)6-s + (−0.226 − 0.974i)8-s + (−0.669 + 0.743i)9-s + (0.458 − 0.888i)12-s + (−0.633 + 0.774i)13-s + (−0.217 + 0.976i)16-s + (0.556 + 0.830i)17-s + (0.924 − 0.380i)18-s + (−0.999 + 0.0380i)19-s + (−0.0950 − 0.995i)23-s + (−0.797 + 0.603i)24-s + (0.905 − 0.424i)26-s + (0.951 + 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.861 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.861 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.861 + 0.508i$
Analytic conductor: \(455.113\)
Root analytic conductor: \(455.113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (1:\ ),\ 0.861 + 0.508i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9676874966 + 0.2641932017i\)
\(L(\frac12)\) \(\approx\) \(0.9676874966 + 0.2641932017i\)
\(L(1)\) \(\approx\) \(0.6045158857 - 0.1556653403i\)
\(L(1)\) \(\approx\) \(0.6045158857 - 0.1556653403i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.901 - 0.432i)T \)
3 \( 1 + (-0.406 - 0.913i)T \)
13 \( 1 + (-0.633 + 0.774i)T \)
17 \( 1 + (0.556 + 0.830i)T \)
19 \( 1 + (-0.999 + 0.0380i)T \)
23 \( 1 + (-0.0950 - 0.995i)T \)
29 \( 1 + (0.696 + 0.717i)T \)
31 \( 1 + (0.710 + 0.703i)T \)
37 \( 1 + (0.836 - 0.548i)T \)
41 \( 1 + (0.941 + 0.336i)T \)
43 \( 1 + (0.755 + 0.654i)T \)
47 \( 1 + (0.924 + 0.380i)T \)
53 \( 1 + (0.976 - 0.217i)T \)
59 \( 1 + (-0.179 + 0.983i)T \)
61 \( 1 + (0.997 + 0.0760i)T \)
67 \( 1 + (-0.971 - 0.235i)T \)
71 \( 1 + (-0.985 - 0.170i)T \)
73 \( 1 + (0.318 - 0.948i)T \)
79 \( 1 + (-0.999 - 0.0190i)T \)
83 \( 1 + (0.113 + 0.993i)T \)
89 \( 1 + (0.928 - 0.371i)T \)
97 \( 1 + (0.389 - 0.921i)T \)
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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

−17.79972314172817042346328470704, −17.430946097806979502770839916678, −16.89393028209681803859806082854, −16.1436661996560599972283204288, −15.540516531429754132599564373777, −15.07223829725590134215927205776, −14.39894405358015938963967545982, −13.5982491366479886620949281594, −12.44393161045631605572713342072, −11.70953691877465119147391961118, −11.19203893996045140100591910675, −10.2305117272176565498925785822, −10.019799261108576781020893751018, −9.256382421357935105564545696989, −8.55285648610730559645683456793, −7.759888382746681771474475988471, −7.10986487214812049077944287896, −6.07088978861169282498296124920, −5.67195744292420333949651763521, −4.85319850855228780472815521679, −4.08543779351952140676275273862, −2.92077852633780948548916066906, −2.3261996204671388365167423031, −0.896713575596737974897242399640, −0.349230904740415412363179048547, 0.71874850099418515896059297706, 1.376404363635233764751827795691, 2.31496478277376149889445338320, 2.7357227237991688793520028133, 3.99705409527016938236104816758, 4.74994991879998576684269475669, 6.05234417354486626303254822322, 6.41810660718389122642326229349, 7.27385382675933859482936176842, 7.815879301322970354667762517817, 8.62930722159169823129443894022, 9.09567298899207006435264664704, 10.338812552920211324904631851979, 10.53469583405421543843508233565, 11.475141155500405487937882809135, 12.11148540223497630009051469888, 12.60474220831766021339264943750, 13.14613937633680418193255080719, 14.2705713885114232153111892303, 14.72193475913623608608236181818, 15.91145468388671391368748070446, 16.58715362430464080327826385190, 16.99093111591068521973104669063, 17.71660021968275939432177191278, 18.235486292263000153047102700350

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