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Properties

Label 1-4235-4235.643-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.0550 - 0.998i$
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  + (−0.996 − 0.0855i)2-s + (0.951 − 0.309i)3-s + (0.985 + 0.170i)4-s + (−0.974 + 0.226i)6-s + (−0.967 − 0.254i)8-s + (0.809 − 0.587i)9-s + (0.989 − 0.142i)12-s + (0.717 − 0.696i)13-s + (0.941 + 0.336i)16-s + (−0.491 − 0.870i)17-s + (−0.856 + 0.516i)18-s + (−0.736 + 0.676i)19-s + (−0.281 − 0.959i)23-s + (−0.998 + 0.0570i)24-s + (−0.774 + 0.633i)26-s + (0.587 − 0.809i)27-s + ⋯
L(s)  = 1  + (−0.996 − 0.0855i)2-s + (0.951 − 0.309i)3-s + (0.985 + 0.170i)4-s + (−0.974 + 0.226i)6-s + (−0.967 − 0.254i)8-s + (0.809 − 0.587i)9-s + (0.989 − 0.142i)12-s + (0.717 − 0.696i)13-s + (0.941 + 0.336i)16-s + (−0.491 − 0.870i)17-s + (−0.856 + 0.516i)18-s + (−0.736 + 0.676i)19-s + (−0.281 − 0.959i)23-s + (−0.998 + 0.0570i)24-s + (−0.774 + 0.633i)26-s + (0.587 − 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0550 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0550 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.0550 - 0.998i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (643, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.0550 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.031346608 - 1.089770259i\)
\(L(\frac12)\) \(\approx\) \(1.031346608 - 1.089770259i\)
\(L(1)\) \(\approx\) \(0.9431821923 - 0.2853359705i\)
\(L(1)\) \(\approx\) \(0.9431821923 - 0.2853359705i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.996 - 0.0855i)T \)
3 \( 1 + (0.951 - 0.309i)T \)
13 \( 1 + (0.717 - 0.696i)T \)
17 \( 1 + (-0.491 - 0.870i)T \)
19 \( 1 + (-0.736 + 0.676i)T \)
23 \( 1 + (-0.281 - 0.959i)T \)
29 \( 1 + (-0.993 + 0.113i)T \)
31 \( 1 + (0.466 + 0.884i)T \)
37 \( 1 + (0.441 - 0.897i)T \)
41 \( 1 + (0.921 + 0.389i)T \)
43 \( 1 + (-0.540 + 0.841i)T \)
47 \( 1 + (0.856 + 0.516i)T \)
53 \( 1 + (0.336 + 0.941i)T \)
59 \( 1 + (-0.921 + 0.389i)T \)
61 \( 1 + (-0.0855 - 0.996i)T \)
67 \( 1 + (0.755 + 0.654i)T \)
71 \( 1 + (0.198 - 0.980i)T \)
73 \( 1 + (0.999 + 0.0285i)T \)
79 \( 1 + (0.362 - 0.931i)T \)
83 \( 1 + (0.791 - 0.610i)T \)
89 \( 1 + (0.415 - 0.909i)T \)
97 \( 1 + (-0.0570 - 0.998i)T \)
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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

−18.76942478982873345128927717738, −17.9499174053143188330285002869, −17.11167545532143376209199808385, −16.63522860497703667885342724951, −15.7292262330940021475027931839, −15.27999655812721125497836669911, −14.813220891910967669954689611508, −13.77489345298292693701945615622, −13.29125001163286262905930379439, −12.38449978925262417218862435541, −11.3693741218010243959108195904, −10.92657530838903300654712233849, −10.12670278314843210727915610424, −9.416613426680582725995030931669, −8.92048187393606662786596926240, −8.27102270329578616734374283758, −7.68552911459559390000092518105, −6.82598232765583992466090680545, −6.22674607726245438770223763100, −5.22387322747880980646317387260, −4.03572212559147142900941104326, −3.62864811867363988328132926188, −2.38759790507661511675780564615, −2.03586715674320430765979912325, −1.05962244941349389741438920790, 0.532766374525267590147434718628, 1.4461730547403552580574263237, 2.26060924626052655659581650778, 2.91598425444295248701915393215, 3.67399161322371726966338349983, 4.57988858745124843639128835565, 5.933584112736172780277049208128, 6.44852884168129380021041990023, 7.352874890129095217902774699519, 7.88129518712921382826183615768, 8.54882325131751739067080342186, 9.112469779117151578840507077468, 9.75457746762616644874075927825, 10.61605368940838279071701655668, 11.05823613843107135634551056374, 12.17693717968153253265752291049, 12.64604135832373966189318157126, 13.38901039155616950146782102479, 14.28990045541805785882800707888, 14.85298315719015913831785203375, 15.64755268592496258360056371198, 16.1004328034836305912516502088, 16.90680726050387102216062128632, 17.78833796219075070330673847348, 18.37423662634273985921282438074

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