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Properties

Label 1-4235-4235.314-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: $\chi_{4235} (314, \cdot )$
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

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

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