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Properties

Label 1-4235-4235.3709-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.920 + 0.391i$
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  + (0.696 − 0.717i)2-s + (0.309 − 0.951i)3-s + (−0.0285 − 0.999i)4-s + (−0.466 − 0.884i)6-s + (−0.736 − 0.676i)8-s + (−0.809 − 0.587i)9-s + (−0.959 − 0.281i)12-s + (−0.610 − 0.791i)13-s + (−0.998 + 0.0570i)16-s + (−0.0855 + 0.996i)17-s + (−0.985 + 0.170i)18-s + (−0.921 + 0.389i)19-s + (−0.841 − 0.540i)23-s + (−0.870 + 0.491i)24-s + (−0.993 − 0.113i)26-s + (−0.809 + 0.587i)27-s + ⋯
L(s)  = 1  + (0.696 − 0.717i)2-s + (0.309 − 0.951i)3-s + (−0.0285 − 0.999i)4-s + (−0.466 − 0.884i)6-s + (−0.736 − 0.676i)8-s + (−0.809 − 0.587i)9-s + (−0.959 − 0.281i)12-s + (−0.610 − 0.791i)13-s + (−0.998 + 0.0570i)16-s + (−0.0855 + 0.996i)17-s + (−0.985 + 0.170i)18-s + (−0.921 + 0.389i)19-s + (−0.841 − 0.540i)23-s + (−0.870 + 0.491i)24-s + (−0.993 − 0.113i)26-s + (−0.809 + 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.920 + 0.391i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (3709, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.920 + 0.391i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2619860159 + 0.05343940920i\)
\(L(\frac12)\) \(\approx\) \(0.2619860159 + 0.05343940920i\)
\(L(1)\) \(\approx\) \(0.8231020297 - 0.8393611646i\)
\(L(1)\) \(\approx\) \(0.8231020297 - 0.8393611646i\)

Euler product

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

−18.03297943692625488967095804548, −17.46290739864623192954494306856, −16.61731905594390254300677641133, −16.280730171167980289400056114621, −15.58605789902622171128755679496, −14.9419920892372763839937573906, −14.31095731111294818755044293699, −13.89712746035722509089386909120, −13.049917190972473217745543077336, −12.30831683839144408923631212876, −11.36379132103624593961350263051, −11.092174666891701446696553989813, −9.77148825449069498750060621519, −9.37329800361982540273989249466, −8.636619356939581167245987309, −7.81076884883042656313509583578, −7.20461218713945736434295081522, −6.2979021142872319480970728539, −5.54643885021746275522371218878, −4.84911634207062813965266880939, −4.189205985496953157733288855875, −3.67686661034668384697914896757, −2.60708486962204601433603881088, −2.13183443899707396529420257352, −0.05266777446692574298223450704, 1.05981394413813739769727406583, 1.97657258094268174777891105367, 2.42670564785626447423550299682, 3.396917508429675175936979505255, 4.00150734713827185172817234391, 5.00428089188143169990840181537, 5.87855045562850462323468161975, 6.292088756789599243563137409353, 7.20737293326508189566337090188, 8.01463934019222028970297671940, 8.70859020758509158619515174483, 9.549501356891357895032679828362, 10.4041587117726886082905143538, 10.928583505283685992486933640163, 11.82969013082570006950401889792, 12.51044954121965357222331100835, 12.83058175166359544551993592751, 13.457300239198557544002670343124, 14.34872381090166584066847591210, 14.8070912936848588557093486894, 15.26135459570488229751219726606, 16.463723546009558785307822495083, 17.18176251891558059692499348911, 18.083916294799763922732375783377, 18.459477532029573319814124738326

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