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Properties

Label 1-3724-3724.3175-r0-0-0
Degree $1$
Conductor $3724$
Sign $0.564 + 0.825i$
Analytic cond. $17.2941$
Root an. cond. $17.2941$
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  + (0.980 + 0.198i)3-s + (−0.661 + 0.749i)5-s + (0.921 + 0.388i)9-s + (−0.955 − 0.294i)11-s + (0.998 − 0.0498i)13-s + (−0.797 + 0.603i)15-s + (0.270 − 0.962i)17-s + (−0.698 + 0.715i)23-s + (−0.124 − 0.992i)25-s + (0.826 + 0.563i)27-s + (−0.995 − 0.0995i)29-s + 31-s + (−0.878 − 0.478i)33-s + (−0.826 + 0.563i)37-s + (0.988 + 0.149i)39-s + ⋯
L(s)  = 1  + (0.980 + 0.198i)3-s + (−0.661 + 0.749i)5-s + (0.921 + 0.388i)9-s + (−0.955 − 0.294i)11-s + (0.998 − 0.0498i)13-s + (−0.797 + 0.603i)15-s + (0.270 − 0.962i)17-s + (−0.698 + 0.715i)23-s + (−0.124 − 0.992i)25-s + (0.826 + 0.563i)27-s + (−0.995 − 0.0995i)29-s + 31-s + (−0.878 − 0.478i)33-s + (−0.826 + 0.563i)37-s + (0.988 + 0.149i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.564 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.564 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $0.564 + 0.825i$
Analytic conductor: \(17.2941\)
Root analytic conductor: \(17.2941\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (3175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3724,\ (0:\ ),\ 0.564 + 0.825i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.920879188 + 1.013008552i\)
\(L(\frac12)\) \(\approx\) \(1.920879188 + 1.013008552i\)
\(L(1)\) \(\approx\) \(1.316767806 + 0.2872485991i\)
\(L(1)\) \(\approx\) \(1.316767806 + 0.2872485991i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.980 + 0.198i)T \)
5 \( 1 + (-0.661 + 0.749i)T \)
11 \( 1 + (-0.955 - 0.294i)T \)
13 \( 1 + (0.998 - 0.0498i)T \)
17 \( 1 + (0.270 - 0.962i)T \)
23 \( 1 + (-0.698 + 0.715i)T \)
29 \( 1 + (-0.995 - 0.0995i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.826 + 0.563i)T \)
41 \( 1 + (0.661 - 0.749i)T \)
43 \( 1 + (0.0249 + 0.999i)T \)
47 \( 1 + (0.998 - 0.0498i)T \)
53 \( 1 + (-0.270 - 0.962i)T \)
59 \( 1 + (0.878 + 0.478i)T \)
61 \( 1 + (0.995 + 0.0995i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (-0.411 + 0.911i)T \)
73 \( 1 + (0.456 + 0.889i)T \)
79 \( 1 + (-0.939 - 0.342i)T \)
83 \( 1 + (0.733 + 0.680i)T \)
89 \( 1 + (0.797 - 0.603i)T \)
97 \( 1 + (0.939 + 0.342i)T \)
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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

−18.724569757067016563873369203851, −17.99634484403711558538601956812, −17.12691611890533151368739518046, −16.25153470199125568104107123286, −15.668603103667674967701915041506, −15.255194616278473640766389655701, −14.38496610786634189020064874215, −13.65128492375453993510519222912, −12.9331019201513442961489230178, −12.56584539384740855080202795832, −11.772235248422025812512784620281, −10.72615030460904233540856974780, −10.19098939823091203573123512150, −9.16746691857149430900194558097, −8.64707290015213991717367772924, −7.960581759655744220665734462939, −7.60106214411865847229855354484, −6.55093944326448370284246329302, −5.68428909469474217536231764804, −4.735865771402949970546117231328, −3.94770945891231342357633424190, −3.48482370070387766101826466937, −2.38640168644344826044446553216, −1.659969283142339378967118715271, −0.66718757147271121637563355104, 0.89719565436823059550605443834, 2.15127612803899106648102098198, 2.81079436510741821106494708806, 3.53154497680068555741096016752, 4.05648907508834360899812139702, 5.08774990003858347425183788404, 5.947671714636976776071988158200, 6.98782567337237163252635515105, 7.53707586369112766787567844183, 8.20110722370883774858521665826, 8.74141611647057757473924625733, 9.803237827895469570249884110551, 10.24377879830634206015242690357, 11.131938145849006806386642474542, 11.628966174157719645839736144706, 12.67786728780275148866804849587, 13.44216162817526854095410288002, 13.93669301635208876753789424640, 14.593169662741246097864254824517, 15.42958253662648157376062719958, 15.875338617577655537083206502772, 16.22866337763986377379419934809, 17.5878646840527171558288486131, 18.2658877146203047648509498948, 18.91691921195534172217387022449

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