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Properties

Label 1-4235-4235.382-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.959 + 0.282i$
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  + (−0.524 + 0.851i)2-s + (0.406 − 0.913i)3-s + (−0.449 − 0.893i)4-s + (0.564 + 0.825i)6-s + (0.996 + 0.0855i)8-s + (−0.669 − 0.743i)9-s + (−0.998 + 0.0475i)12-s + (0.967 − 0.254i)13-s + (−0.595 + 0.803i)16-s + (0.768 − 0.640i)17-s + (0.983 − 0.179i)18-s + (−0.969 + 0.244i)19-s + (0.814 − 0.580i)23-s + (0.483 − 0.875i)24-s + (−0.290 + 0.956i)26-s + (−0.951 + 0.309i)27-s + ⋯
L(s)  = 1  + (−0.524 + 0.851i)2-s + (0.406 − 0.913i)3-s + (−0.449 − 0.893i)4-s + (0.564 + 0.825i)6-s + (0.996 + 0.0855i)8-s + (−0.669 − 0.743i)9-s + (−0.998 + 0.0475i)12-s + (0.967 − 0.254i)13-s + (−0.595 + 0.803i)16-s + (0.768 − 0.640i)17-s + (0.983 − 0.179i)18-s + (−0.969 + 0.244i)19-s + (0.814 − 0.580i)23-s + (0.483 − 0.875i)24-s + (−0.290 + 0.956i)26-s + (−0.951 + 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.959 + 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.959 + 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.959 + 0.282i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (382, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.959 + 0.282i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001371324029 + 0.009520451458i\)
\(L(\frac12)\) \(\approx\) \(0.001371324029 + 0.009520451458i\)
\(L(1)\) \(\approx\) \(0.7535475521 + 0.01563986513i\)
\(L(1)\) \(\approx\) \(0.7535475521 + 0.01563986513i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.524 + 0.851i)T \)
3 \( 1 + (0.406 - 0.913i)T \)
13 \( 1 + (0.967 - 0.254i)T \)
17 \( 1 + (0.768 - 0.640i)T \)
19 \( 1 + (-0.969 + 0.244i)T \)
23 \( 1 + (0.814 - 0.580i)T \)
29 \( 1 + (-0.466 + 0.884i)T \)
31 \( 1 + (-0.935 - 0.353i)T \)
37 \( 1 + (-0.151 + 0.988i)T \)
41 \( 1 + (-0.610 + 0.791i)T \)
43 \( 1 + (-0.755 - 0.654i)T \)
47 \( 1 + (-0.983 - 0.179i)T \)
53 \( 1 + (-0.803 + 0.595i)T \)
59 \( 1 + (-0.380 + 0.924i)T \)
61 \( 1 + (-0.879 - 0.475i)T \)
67 \( 1 + (-0.690 + 0.723i)T \)
71 \( 1 + (0.897 - 0.441i)T \)
73 \( 1 + (-0.508 + 0.861i)T \)
79 \( 1 + (0.123 - 0.992i)T \)
83 \( 1 + (-0.676 - 0.736i)T \)
89 \( 1 + (0.786 + 0.618i)T \)
97 \( 1 + (-0.856 + 0.516i)T \)
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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

−18.97232051562048021557106894319, −18.17836527322687079768142154142, −17.29919926233693657794109416023, −16.80822407015565735422622005533, −16.18693235993113605180607644262, −15.40191679866286522313043245935, −14.69791124291997081035206322085, −13.92511056727812427401759626518, −13.22111644863803451236456801188, −12.629425971784417830012977333346, −11.65115977093081136707167473304, −10.91798390768798384202733576255, −10.698199819442248107661942532, −9.6969321374764644185272830109, −9.28415801852279487457771324290, −8.48018439468889363386699065424, −8.06990073029384934887037816598, −7.11595613724374014448561374303, −6.025044771610431875388873525332, −5.132157654555600196281469047750, −4.32241972877572605008936966753, −3.59154016480934973259673860216, −3.20173337907043400871829841090, −2.085750177239107526067204870265, −1.4716415450953117080164359154, 0.00301277948479895691142498537, 1.197345621657773176859142691354, 1.68181969234348841022728262183, 2.88213292319373613007263857830, 3.66197923415223721247574873764, 4.795058262441367496859574121, 5.579875672834512253577511835850, 6.3335043833077883214757099120, 6.84128824358052087325397151938, 7.60891593800440950858570944304, 8.23799551675137349715329232337, 8.80550970865629506327386428912, 9.41527228587135953652242850421, 10.380628412898978320738396278952, 11.03613405940973818758332745218, 11.90198099932924885545958064441, 12.85902437448954080488404829721, 13.31299267913276509875514629017, 14.00123271030658812856633933773, 14.8356936701765189957295551646, 15.033054189664985945533269052680, 16.13777919781497661360884368550, 16.726274828332091732162063243992, 17.29680611294494613184431960881, 18.189581216184188155096394156208

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