↑

Properties

Label 1-4235-4235.39-r1-0-0
Degree $1$
Conductor $4235$
Sign $-0.675 - 0.737i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.948 − 0.318i)2-s + (−0.669 + 0.743i)3-s + (0.797 + 0.603i)4-s + (0.870 − 0.491i)6-s + (−0.564 − 0.825i)8-s + (−0.104 − 0.994i)9-s + (−0.981 + 0.189i)12-s + (0.974 + 0.226i)13-s + (0.272 + 0.962i)16-s + (−0.625 + 0.780i)17-s + (−0.217 + 0.976i)18-s + (0.935 − 0.353i)19-s + (0.786 + 0.618i)23-s + (0.991 + 0.132i)24-s + (−0.851 − 0.524i)26-s + (0.809 + 0.587i)27-s + ⋯
L(s)  = 1  + (−0.948 − 0.318i)2-s + (−0.669 + 0.743i)3-s + (0.797 + 0.603i)4-s + (0.870 − 0.491i)6-s + (−0.564 − 0.825i)8-s + (−0.104 − 0.994i)9-s + (−0.981 + 0.189i)12-s + (0.974 + 0.226i)13-s + (0.272 + 0.962i)16-s + (−0.625 + 0.780i)17-s + (−0.217 + 0.976i)18-s + (0.935 − 0.353i)19-s + (0.786 + 0.618i)23-s + (0.991 + 0.132i)24-s + (−0.851 − 0.524i)26-s + (0.809 + 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.675 - 0.737i$
Analytic conductor: \(455.113\)
Root analytic conductor: \(455.113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (1:\ ),\ -0.675 - 0.737i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04146342869 + 0.09413825664i\)
\(L(\frac12)\) \(\approx\) \(-0.04146342869 + 0.09413825664i\)
\(L(1)\) \(\approx\) \(0.5417646114 + 0.1184328273i\)
\(L(1)\) \(\approx\) \(0.5417646114 + 0.1184328273i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.948 - 0.318i)T \)
3 \( 1 + (-0.669 + 0.743i)T \)
13 \( 1 + (0.974 + 0.226i)T \)
17 \( 1 + (-0.625 + 0.780i)T \)
19 \( 1 + (0.935 - 0.353i)T \)
23 \( 1 + (0.786 + 0.618i)T \)
29 \( 1 + (0.254 + 0.967i)T \)
31 \( 1 + (-0.905 + 0.424i)T \)
37 \( 1 + (0.999 - 0.0190i)T \)
41 \( 1 + (-0.993 - 0.113i)T \)
43 \( 1 + (-0.959 + 0.281i)T \)
47 \( 1 + (0.217 + 0.976i)T \)
53 \( 1 + (-0.272 + 0.962i)T \)
59 \( 1 + (-0.595 - 0.803i)T \)
61 \( 1 + (-0.749 - 0.662i)T \)
67 \( 1 + (0.995 - 0.0950i)T \)
71 \( 1 + (-0.998 - 0.0570i)T \)
73 \( 1 + (-0.0665 - 0.997i)T \)
79 \( 1 + (0.179 - 0.983i)T \)
83 \( 1 + (-0.466 + 0.884i)T \)
89 \( 1 + (-0.888 + 0.458i)T \)
97 \( 1 + (-0.610 + 0.791i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.011306072186065944234629915814, −17.04687924777880647868879732373, −16.652777359423615693096074001402, −15.948252194730466471390473736589, −15.34111885697237959361593849946, −14.44911870668547591070103881146, −13.57270818099590351203307433583, −13.10843339386969711713502525870, −12.024248894774828520597371485428, −11.498772376384217285207336465043, −10.99298996667453263262598953354, −10.21482012473013646827270295604, −9.48878337248039765979813127999, −8.57019171830163076247928478537, −8.09742181662737906678665214525, −7.17614190754350643137425202431, −6.81323343690765502727183043871, −5.92550580689437679344501049354, −5.447446372202905537699827668940, −4.52076613905388703755677740498, −3.188306267325136571861793954285, −2.3454922492522806875103285984, −1.48508679271315786322688090794, −0.76142165999420775216771370338, −0.03151200221988449281874609064, 1.09060689701785024620014961125, 1.65037162969593293431534992697, 3.054936082867994776937652358818, 3.47449003062509441481581126938, 4.40863928786696897750971643634, 5.29536530038494019118179216003, 6.183041925123459136971576198, 6.72653357381990740380974658679, 7.56925233063262579614326250694, 8.53135149869674223021759522809, 9.11558631907927370732406746557, 9.6199653396365822849982225054, 10.527017689931576516424746794245, 11.054432073461744113874165430697, 11.40306943272874178465574123692, 12.316663550264555404169112620902, 12.93223503309860480488833108182, 13.84866020034284872434089398595, 14.95787921824000083956559354583, 15.47559951635317280364745918205, 16.10383577271105556488184044600, 16.65820869805016758905615620026, 17.290334579053754802238634769855, 18.01238581335993497463939001039, 18.36230265162246886749859943467

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