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Properties

Label 1-4235-4235.2953-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.576 + 0.816i$
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  + (0.717 − 0.696i)2-s + (0.951 − 0.309i)3-s + (0.0285 − 0.999i)4-s + (0.466 − 0.884i)6-s + (−0.676 − 0.736i)8-s + (0.809 − 0.587i)9-s + (−0.281 − 0.959i)12-s + (−0.791 − 0.610i)13-s + (−0.998 − 0.0570i)16-s + (−0.996 + 0.0855i)17-s + (0.170 − 0.985i)18-s + (−0.921 − 0.389i)19-s + (0.540 + 0.841i)23-s + (−0.870 − 0.491i)24-s + (−0.993 + 0.113i)26-s + (0.587 − 0.809i)27-s + ⋯
L(s)  = 1  + (0.717 − 0.696i)2-s + (0.951 − 0.309i)3-s + (0.0285 − 0.999i)4-s + (0.466 − 0.884i)6-s + (−0.676 − 0.736i)8-s + (0.809 − 0.587i)9-s + (−0.281 − 0.959i)12-s + (−0.791 − 0.610i)13-s + (−0.998 − 0.0570i)16-s + (−0.996 + 0.0855i)17-s + (0.170 − 0.985i)18-s + (−0.921 − 0.389i)19-s + (0.540 + 0.841i)23-s + (−0.870 − 0.491i)24-s + (−0.993 + 0.113i)26-s + (0.587 − 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.576 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.576 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.576 + 0.816i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (2953, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.576 + 0.816i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.6518167701 - 1.258145012i\)
\(L(\frac12)\) \(\approx\) \(-0.6518167701 - 1.258145012i\)
\(L(1)\) \(\approx\) \(1.170120894 - 1.041080865i\)
\(L(1)\) \(\approx\) \(1.170120894 - 1.041080865i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.717 - 0.696i)T \)
3 \( 1 + (0.951 - 0.309i)T \)
13 \( 1 + (-0.791 - 0.610i)T \)
17 \( 1 + (-0.996 + 0.0855i)T \)
19 \( 1 + (-0.921 - 0.389i)T \)
23 \( 1 + (0.540 + 0.841i)T \)
29 \( 1 + (-0.516 - 0.856i)T \)
31 \( 1 + (-0.941 + 0.336i)T \)
37 \( 1 + (0.825 + 0.564i)T \)
41 \( 1 + (-0.897 + 0.441i)T \)
43 \( 1 + (-0.909 + 0.415i)T \)
47 \( 1 + (-0.170 - 0.985i)T \)
53 \( 1 + (-0.0570 - 0.998i)T \)
59 \( 1 + (0.897 + 0.441i)T \)
61 \( 1 + (-0.696 + 0.717i)T \)
67 \( 1 + (-0.989 + 0.142i)T \)
71 \( 1 + (0.974 - 0.226i)T \)
73 \( 1 + (-0.967 + 0.254i)T \)
79 \( 1 + (-0.198 - 0.980i)T \)
83 \( 1 + (-0.931 + 0.362i)T \)
89 \( 1 + (-0.654 - 0.755i)T \)
97 \( 1 + (0.491 - 0.870i)T \)
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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

−18.73934497972301098991169703613, −18.2092382041569014247937776142, −17.07529999588357098093466855852, −16.71682959194727735929974913929, −15.96726485057472585608459174541, −15.23665670758489348591637256374, −14.700775892892750716416546889012, −14.3015356319235906052503166336, −13.49469363940971377018599721350, −12.83751823866296836403356104205, −12.39553775466288869141589005943, −11.293337301008378121150082630299, −10.665213468091533463857522668045, −9.61610488918867265626611176594, −8.92587007707425264194688890620, −8.48537047398475348057586771498, −7.56514494227345448697441137826, −7.01732169663690350652608260900, −6.3509490846968937853446164892, −5.284430476321480296673624231366, −4.55526495836225356665558662461, −4.08951822988467331795065328723, −3.23060028039822492899706413538, −2.40940330362152608794070330425, −1.83125722892262774507812762963, 0.21780904195999123859814174138, 1.48520457358326188879337223644, 2.13326397332379694690529013670, 2.83822633657259814262996984292, 3.511494611422546958689683560045, 4.34082783458052632202203655969, 4.96596913968495756562889104989, 5.93195628950086746667196103614, 6.78007295590595092642923551026, 7.349440960036063317255293422253, 8.37762319279374130011411687463, 9.00194558418021965258470031152, 9.7959646808194229275776442827, 10.286281420076808024826966186632, 11.26838240163866744803651955199, 11.808722376222287095929235504127, 12.81222040052738213540877806688, 13.1474930212919353935335347536, 13.59901983719760830553014242845, 14.619857015456612427719162375748, 15.08959911531599911208678554512, 15.35501861303352632870693861899, 16.47217765957447451309794077583, 17.47901693158982563909812323526, 18.139296961655775331303073474102

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