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Properties

Label 1-4235-4235.767-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.998 + 0.0627i$
Analytic cond. $19.6672$
Root an. cond. $19.6672$
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  + (0.263 − 0.964i)2-s + (0.406 − 0.913i)3-s + (−0.861 − 0.508i)4-s + (−0.774 − 0.633i)6-s + (−0.717 + 0.696i)8-s + (−0.669 − 0.743i)9-s + (−0.814 + 0.580i)12-s + (0.676 − 0.736i)13-s + (0.483 + 0.875i)16-s + (0.524 + 0.851i)17-s + (−0.893 + 0.449i)18-s + (0.380 + 0.924i)19-s + (−0.189 − 0.981i)23-s + (0.345 + 0.938i)24-s + (−0.532 − 0.846i)26-s + (−0.951 + 0.309i)27-s + ⋯
L(s)  = 1  + (0.263 − 0.964i)2-s + (0.406 − 0.913i)3-s + (−0.861 − 0.508i)4-s + (−0.774 − 0.633i)6-s + (−0.717 + 0.696i)8-s + (−0.669 − 0.743i)9-s + (−0.814 + 0.580i)12-s + (0.676 − 0.736i)13-s + (0.483 + 0.875i)16-s + (0.524 + 0.851i)17-s + (−0.893 + 0.449i)18-s + (0.380 + 0.924i)19-s + (−0.189 − 0.981i)23-s + (0.345 + 0.938i)24-s + (−0.532 − 0.846i)26-s + (−0.951 + 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.998 + 0.0627i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (767, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.998 + 0.0627i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06236390085 - 1.984667999i\)
\(L(\frac12)\) \(\approx\) \(-0.06236390085 - 1.984667999i\)
\(L(1)\) \(\approx\) \(0.7793102971 - 1.005663469i\)
\(L(1)\) \(\approx\) \(0.7793102971 - 1.005663469i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.263 - 0.964i)T \)
3 \( 1 + (0.406 - 0.913i)T \)
13 \( 1 + (0.676 - 0.736i)T \)
17 \( 1 + (0.524 + 0.851i)T \)
19 \( 1 + (0.380 + 0.924i)T \)
23 \( 1 + (-0.189 - 0.981i)T \)
29 \( 1 + (0.941 + 0.336i)T \)
31 \( 1 + (-0.595 - 0.803i)T \)
37 \( 1 + (-0.662 + 0.749i)T \)
41 \( 1 + (0.362 - 0.931i)T \)
43 \( 1 + (0.989 - 0.142i)T \)
47 \( 1 + (0.893 + 0.449i)T \)
53 \( 1 + (-0.875 - 0.483i)T \)
59 \( 1 + (0.625 - 0.780i)T \)
61 \( 1 + (0.710 + 0.703i)T \)
67 \( 1 + (0.458 - 0.888i)T \)
71 \( 1 + (-0.564 - 0.825i)T \)
73 \( 1 + (0.572 + 0.820i)T \)
79 \( 1 + (-0.830 + 0.556i)T \)
83 \( 1 + (0.389 - 0.921i)T \)
89 \( 1 + (-0.723 + 0.690i)T \)
97 \( 1 + (0.170 - 0.985i)T \)
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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

−18.64775888295725394481913840415, −17.75054062830422896340004529242, −17.3113532206163034148385155898, −16.262468231237285342125698560246, −16.04181298709009325747949935815, −15.52685174921172871415818385221, −14.65997884923904605049135776934, −13.982474828720610030392885053047, −13.75500407692739976694540267966, −12.83328166262889781553892460691, −11.84241129833026962632763942047, −11.24607320026449528950412313249, −10.28899042167731049407714582186, −9.45092494627743242215768170491, −9.085244945875700962935248892651, −8.389397823711237695637512533525, −7.541162816998683202770790586715, −6.95844681449025730236440257243, −5.96540128446801884525365715790, −5.30791262032872822350337856674, −4.660372125573938846677641725408, −3.92353355470766212114935489006, −3.26961853810402660005493770429, −2.45742665339189107690703342595, −1.007652355376224279369322134939, 0.56568234921285174350535004265, 1.351150453743432120612110571634, 2.06830823002480998501599939524, 2.92967182453066999826675843094, 3.55995281188848608246945910735, 4.25698371924616806057609847397, 5.515245326117629087526158244672, 5.89810780523705073223604773153, 6.76940256036884789603796438584, 7.89838946475286466087633730731, 8.30656353655027532002748219935, 8.99927250914363293155204302222, 9.91329302465932769862487551348, 10.558027396640609002890901413757, 11.23010109400683634433599450117, 12.19262714187728992623631140479, 12.504284628419954206715929216001, 13.07930420881930200039044385877, 13.94967386677304017425170295688, 14.30717007174155851206578475940, 15.02750837017581729644442872191, 15.878863853365166606738020201045, 16.972406564504581926379220209638, 17.6305259188501809207178990725, 18.23953785254549469567411707511

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