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Properties

Label 1-4235-4235.1119-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.261 - 0.965i$
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  + (0.993 − 0.113i)2-s + (−0.809 + 0.587i)3-s + (0.974 − 0.226i)4-s + (−0.736 + 0.676i)6-s + (0.941 − 0.336i)8-s + (0.309 − 0.951i)9-s + (−0.654 + 0.755i)12-s + (−0.516 − 0.856i)13-s + (0.897 − 0.441i)16-s + (−0.774 − 0.633i)17-s + (0.198 − 0.980i)18-s + (−0.998 + 0.0570i)19-s + (0.142 + 0.989i)23-s + (−0.564 + 0.825i)24-s + (−0.610 − 0.791i)26-s + (0.309 + 0.951i)27-s + ⋯
L(s)  = 1  + (0.993 − 0.113i)2-s + (−0.809 + 0.587i)3-s + (0.974 − 0.226i)4-s + (−0.736 + 0.676i)6-s + (0.941 − 0.336i)8-s + (0.309 − 0.951i)9-s + (−0.654 + 0.755i)12-s + (−0.516 − 0.856i)13-s + (0.897 − 0.441i)16-s + (−0.774 − 0.633i)17-s + (0.198 − 0.980i)18-s + (−0.998 + 0.0570i)19-s + (0.142 + 0.989i)23-s + (−0.564 + 0.825i)24-s + (−0.610 − 0.791i)26-s + (0.309 + 0.951i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.261 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.261 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.261 - 0.965i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.261 - 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9266024673 - 1.211366209i\)
\(L(\frac12)\) \(\approx\) \(0.9266024673 - 1.211366209i\)
\(L(1)\) \(\approx\) \(1.320838163 - 0.1346752822i\)
\(L(1)\) \(\approx\) \(1.320838163 - 0.1346752822i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.993 - 0.113i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
13 \( 1 + (-0.516 - 0.856i)T \)
17 \( 1 + (-0.774 - 0.633i)T \)
19 \( 1 + (-0.998 + 0.0570i)T \)
23 \( 1 + (0.142 + 0.989i)T \)
29 \( 1 + (0.362 + 0.931i)T \)
31 \( 1 + (0.921 - 0.389i)T \)
37 \( 1 + (-0.0855 - 0.996i)T \)
41 \( 1 + (-0.870 - 0.491i)T \)
43 \( 1 + (-0.959 - 0.281i)T \)
47 \( 1 + (0.198 + 0.980i)T \)
53 \( 1 + (-0.897 - 0.441i)T \)
59 \( 1 + (0.870 - 0.491i)T \)
61 \( 1 + (0.993 + 0.113i)T \)
67 \( 1 + (-0.415 - 0.909i)T \)
71 \( 1 + (-0.254 + 0.967i)T \)
73 \( 1 + (0.466 - 0.884i)T \)
79 \( 1 + (0.0285 - 0.999i)T \)
83 \( 1 + (0.985 - 0.170i)T \)
89 \( 1 + (-0.841 + 0.540i)T \)
97 \( 1 + (-0.564 + 0.825i)T \)
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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

−18.67886324035988190291749467553, −17.69027443884513775444939082992, −16.9368499377838183651015996869, −16.76209805522087709483871144807, −15.79068388526070043127424001999, −15.153275861335933442113767501779, −14.452430982187659780727824810613, −13.61434301170066957426010001442, −13.159007307290138243417186082749, −12.46734677945627356450496888935, −11.813337272416290542660112591908, −11.38025155131198921633290471855, −10.50426054748315739259474883583, −10.01376921629746523957542007379, −8.547492377979878975578030522916, −8.09231428921764629112421403490, −6.95465349810402210003588892775, −6.592155076287870320024921647392, −6.13308799603490776000623223035, −5.03508629862147901247920306628, −4.60029083383278643285124455982, −3.91241732621202868954509650875, −2.60684228050771924188831435450, −2.11646587159408488887182578344, −1.20801278416991992350249355198, 0.31045815532385360508323336730, 1.508100451505739450783315301571, 2.51755269948914029453311308550, 3.3427243916911264092874376594, 4.05568329903550391811769113970, 4.91451339035392070263278752829, 5.23290428884296831145240002787, 6.124689136769075324514581574380, 6.742988296043441950983131267688, 7.430720669740616580789259867922, 8.46415397208341717840817083439, 9.4902156084003722997079308058, 10.16697766061278985561285115143, 10.81062417715658343799283124969, 11.35339573451962940297027669900, 12.08042591094506407594756416175, 12.6710031099627687741197650924, 13.29956954283237278274466827410, 14.13818540193034782138114818693, 14.96572112124407733979457407360, 15.39010534364122727830842606511, 15.990266124938515284254261563576, 16.66972631675645310588272542784, 17.469817758398997048091432699870, 17.86418781014705322758393744000

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