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Properties

Label 1-4235-4235.97-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.988 + 0.150i$
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  + (−0.884 − 0.466i)2-s + (0.951 + 0.309i)3-s + (0.564 + 0.825i)4-s + (−0.696 − 0.717i)6-s + (−0.113 − 0.993i)8-s + (0.809 + 0.587i)9-s + (0.281 + 0.959i)12-s + (−0.336 + 0.941i)13-s + (−0.362 + 0.931i)16-s + (0.226 − 0.974i)17-s + (−0.441 − 0.897i)18-s + (0.516 + 0.856i)19-s + (−0.540 − 0.841i)23-s + (0.198 − 0.980i)24-s + (0.736 − 0.676i)26-s + (0.587 + 0.809i)27-s + ⋯
L(s)  = 1  + (−0.884 − 0.466i)2-s + (0.951 + 0.309i)3-s + (0.564 + 0.825i)4-s + (−0.696 − 0.717i)6-s + (−0.113 − 0.993i)8-s + (0.809 + 0.587i)9-s + (0.281 + 0.959i)12-s + (−0.336 + 0.941i)13-s + (−0.362 + 0.931i)16-s + (0.226 − 0.974i)17-s + (−0.441 − 0.897i)18-s + (0.516 + 0.856i)19-s + (−0.540 − 0.841i)23-s + (0.198 − 0.980i)24-s + (0.736 − 0.676i)26-s + (0.587 + 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.988 + 0.150i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.988 + 0.150i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.780958586 + 0.1348428754i\)
\(L(\frac12)\) \(\approx\) \(1.780958586 + 0.1348428754i\)
\(L(1)\) \(\approx\) \(1.072004575 + 0.01203637037i\)
\(L(1)\) \(\approx\) \(1.072004575 + 0.01203637037i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.884 - 0.466i)T \)
3 \( 1 + (0.951 + 0.309i)T \)
13 \( 1 + (-0.336 + 0.941i)T \)
17 \( 1 + (0.226 - 0.974i)T \)
19 \( 1 + (0.516 + 0.856i)T \)
23 \( 1 + (-0.540 - 0.841i)T \)
29 \( 1 + (0.921 + 0.389i)T \)
31 \( 1 + (-0.610 - 0.791i)T \)
37 \( 1 + (0.999 - 0.0285i)T \)
41 \( 1 + (0.985 + 0.170i)T \)
43 \( 1 + (0.909 - 0.415i)T \)
47 \( 1 + (0.441 - 0.897i)T \)
53 \( 1 + (-0.931 + 0.362i)T \)
59 \( 1 + (-0.985 + 0.170i)T \)
61 \( 1 + (0.466 + 0.884i)T \)
67 \( 1 + (0.989 - 0.142i)T \)
71 \( 1 + (0.0855 - 0.996i)T \)
73 \( 1 + (-0.633 + 0.774i)T \)
79 \( 1 + (0.870 - 0.491i)T \)
83 \( 1 + (-0.0570 - 0.998i)T \)
89 \( 1 + (-0.654 - 0.755i)T \)
97 \( 1 + (-0.980 - 0.198i)T \)
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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

−18.20511611127299600636361938610, −17.75273828903709070439616306947, −17.27272707854748433635301707297, −16.17777564711466390364290802173, −15.59993535472831350581815982858, −15.1653566587442402069018268024, −14.29752656009261889082284778754, −13.94195636281198806063357279216, −12.858369826664452067377520588449, −12.40320787269147181131337156508, −11.29951598482704916367159888926, −10.64312804548002861016848545194, −9.76479088188418158173635787239, −9.42244818704869635081325267065, −8.55586292177465954591109420050, −7.87079912019688900308513592468, −7.568072498355008041557753625641, −6.649321489414515788709703544255, −5.95856162884246829130391713436, −5.10796541520566133238965738827, −4.08876295624071951857719814251, −3.04820915962649485484508633636, −2.47329362767994022703520406907, −1.503817779548705775527816753802, −0.774919481882442271564788128802, 0.82903471652671280621897303202, 1.83868253433018962562026646388, 2.47954450941910303546755564978, 3.15387615271283840804329660893, 4.062735870840069069510947757346, 4.59168454970732112895194239822, 5.87611459102250751470451868791, 6.870296958337689845405151300262, 7.52421618901924678535999404214, 8.04633404228854897628923621601, 8.91277500879482377932544304786, 9.39878984907907484134395479048, 9.95925161366592786946599035027, 10.64984079552404298903088255868, 11.47031508944910379692733568840, 12.19825529863511628609900818962, 12.784548143702702575862144212620, 13.80065154308129183328249332633, 14.24803257288372693397528529797, 15.02462659315086388674520886300, 16.00902552115832926059269836772, 16.289232968517892186361725068422, 16.93391390815769130549963923151, 18.00580110362167058478042268965, 18.5276061206886971906881238082

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