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Properties

Label 1-3784-3784.971-r1-0-0
Degree $1$
Conductor $3784$
Sign $-0.514 - 0.857i$
Analytic cond. $406.647$
Root an. cond. $406.647$
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  + (−0.842 − 0.538i)3-s + (−0.971 + 0.237i)5-s + (0.104 − 0.994i)7-s + (0.420 + 0.907i)9-s + (−0.791 + 0.611i)13-s + (0.946 + 0.323i)15-s + (−0.925 + 0.379i)17-s + (0.992 + 0.119i)19-s + (−0.623 + 0.781i)21-s + (−0.955 − 0.294i)23-s + (0.887 − 0.460i)25-s + (0.134 − 0.990i)27-s + (0.842 − 0.538i)29-s + (0.163 − 0.986i)31-s + (0.134 + 0.990i)35-s + ⋯
L(s)  = 1  + (−0.842 − 0.538i)3-s + (−0.971 + 0.237i)5-s + (0.104 − 0.994i)7-s + (0.420 + 0.907i)9-s + (−0.791 + 0.611i)13-s + (0.946 + 0.323i)15-s + (−0.925 + 0.379i)17-s + (0.992 + 0.119i)19-s + (−0.623 + 0.781i)21-s + (−0.955 − 0.294i)23-s + (0.887 − 0.460i)25-s + (0.134 − 0.990i)27-s + (0.842 − 0.538i)29-s + (0.163 − 0.986i)31-s + (0.134 + 0.990i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3784 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.514 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3784 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.514 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3784\)    =    \(2^{3} \cdot 11 \cdot 43\)
Sign: $-0.514 - 0.857i$
Analytic conductor: \(406.647\)
Root analytic conductor: \(406.647\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3784} (971, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3784,\ (1:\ ),\ -0.514 - 0.857i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2046857248 - 0.3617064596i\)
\(L(\frac12)\) \(\approx\) \(0.2046857248 - 0.3617064596i\)
\(L(1)\) \(\approx\) \(0.5781995079 - 0.1017234388i\)
\(L(1)\) \(\approx\) \(0.5781995079 - 0.1017234388i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
43 \( 1 \)
good3 \( 1 + (-0.842 - 0.538i)T \)
5 \( 1 + (-0.971 + 0.237i)T \)
7 \( 1 + (0.104 - 0.994i)T \)
13 \( 1 + (-0.791 + 0.611i)T \)
17 \( 1 + (-0.925 + 0.379i)T \)
19 \( 1 + (0.992 + 0.119i)T \)
23 \( 1 + (-0.955 - 0.294i)T \)
29 \( 1 + (0.842 - 0.538i)T \)
31 \( 1 + (0.163 - 0.986i)T \)
37 \( 1 + (-0.913 + 0.406i)T \)
41 \( 1 + (-0.963 - 0.266i)T \)
47 \( 1 + (0.393 + 0.919i)T \)
53 \( 1 + (0.280 + 0.959i)T \)
59 \( 1 + (0.473 + 0.880i)T \)
61 \( 1 + (0.163 + 0.986i)T \)
67 \( 1 + (-0.733 - 0.680i)T \)
71 \( 1 + (-0.575 - 0.817i)T \)
73 \( 1 + (-0.337 - 0.941i)T \)
79 \( 1 + (0.978 - 0.207i)T \)
83 \( 1 + (-0.772 + 0.635i)T \)
89 \( 1 + (0.365 + 0.930i)T \)
97 \( 1 + (-0.995 - 0.0896i)T \)
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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

−18.507337646668355673145384350556, −17.82926583756362047515602423847, −17.46407924998856307248062755081, −16.312311460192511708631656357081, −15.92791313028050557809667246155, −15.45453600192193702025775046694, −14.835124861764612528320624930678, −13.955409297324460985489054648020, −12.7944246813543149796752728902, −12.26402760822894848191765111476, −11.7342278485632442186225522800, −11.27301378922060536871578765537, −10.30246767411379468363444140869, −9.72314338161657058331197538316, −8.78941990500314781853765241351, −8.31644051807144156987743746472, −7.21134597722427330010741283172, −6.72325477663689078344441008684, −5.56005172795866755181992205039, −5.13557713164422542141736696436, −4.51904064225442716963191504671, −3.53448279255902774318321017345, −2.8739321026857207724812921632, −1.679874994094572317633502072369, −0.50356745398201013795568655146, 0.150149859079848868771357712286, 0.96664850301297043738069156718, 1.93248061750892294571832103317, 2.91058466549919233015164280240, 4.16734037313262724547059664053, 4.3521378917472146264898120886, 5.30571126250357434609969071870, 6.40778999414651749978267982561, 6.838029909012067278400873179447, 7.65131983092455834784664249862, 7.950016915402571202933254009314, 9.08439434959794747690242677035, 10.21829059412635548645889664246, 10.54096636665736118071881468875, 11.46019057460052571700197185093, 11.913203716910211324154740441970, 12.413918522193103091038579281622, 13.59375520973059929414310370310, 13.76643403518051843607366492465, 14.82972515804780614333913396183, 15.58171519584300567662441374067, 16.3169926464259236811115543198, 16.78581396722252172472443125209, 17.56578121292010241765620755316, 18.06688400010238067294965685802

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