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Properties

Label 2-3808-3808.2141-c0-0-6
Degree $2$
Conductor $3808$
Sign $0.831 - 0.555i$
Analytic cond. $1.90043$
Root an. cond. $1.37856$
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  + (0.156 + 0.987i)2-s + (0.431 + 0.178i)3-s + (−0.951 + 0.309i)4-s + (−0.0600 − 0.144i)5-s + (−0.108 + 0.453i)6-s + (0.707 − 0.707i)7-s + (−0.453 − 0.891i)8-s + (−0.552 − 0.552i)9-s + (0.133 − 0.0819i)10-s + (−0.465 − 0.0366i)12-s + (0.809 + 0.587i)14-s − 0.0732i·15-s + (0.809 − 0.587i)16-s + i·17-s + (0.459 − 0.632i)18-s + ⋯
L(s)  = 1  + (0.156 + 0.987i)2-s + (0.431 + 0.178i)3-s + (−0.951 + 0.309i)4-s + (−0.0600 − 0.144i)5-s + (−0.108 + 0.453i)6-s + (0.707 − 0.707i)7-s + (−0.453 − 0.891i)8-s + (−0.552 − 0.552i)9-s + (0.133 − 0.0819i)10-s + (−0.465 − 0.0366i)12-s + (0.809 + 0.587i)14-s − 0.0732i·15-s + (0.809 − 0.587i)16-s + i·17-s + (0.459 − 0.632i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3808\)    =    \(2^{5} \cdot 7 \cdot 17\)
Sign: $0.831 - 0.555i$
Analytic conductor: \(1.90043\)
Root analytic conductor: \(1.37856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3808} (2141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3808,\ (\ :0),\ 0.831 - 0.555i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.457066281\)
\(L(\frac12)\) \(\approx\) \(1.457066281\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.156 - 0.987i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
17 \( 1 - iT \)
good3 \( 1 + (-0.431 - 0.178i)T + (0.707 + 0.707i)T^{2} \)
5 \( 1 + (0.0600 + 0.144i)T + (-0.707 + 0.707i)T^{2} \)
11 \( 1 + (-0.707 + 0.707i)T^{2} \)
13 \( 1 + (0.707 + 0.707i)T^{2} \)
19 \( 1 + (0.707 + 0.707i)T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + (-0.707 - 0.707i)T^{2} \)
31 \( 1 - 0.618T + T^{2} \)
37 \( 1 + (0.707 - 0.707i)T^{2} \)
41 \( 1 + (1.14 + 1.14i)T + iT^{2} \)
43 \( 1 + (-1.57 + 0.652i)T + (0.707 - 0.707i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-1.40 + 0.581i)T + (0.707 - 0.707i)T^{2} \)
59 \( 1 + (0.707 - 0.707i)T^{2} \)
61 \( 1 + (-1.79 - 0.744i)T + (0.707 + 0.707i)T^{2} \)
67 \( 1 + (-1.20 - 0.497i)T + (0.707 + 0.707i)T^{2} \)
71 \( 1 + iT^{2} \)
73 \( 1 + (-0.221 - 0.221i)T + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (0.707 + 0.707i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 + 0.907T + T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.409394323907232438414931847669, −8.256584426313307907552888885685, −7.21447583489220768547407785535, −6.65833715354765458922270208680, −5.75172627033416960371049091879, −5.08134272166875999838634019705, −4.03595801478624944567091455210, −3.77601627187836899467480544568, −2.46662153128984277658460740372, −0.876310951112494298634045757971, 1.25061755204699369555888629357, 2.40524743672256137011116841790, 2.77091021599380360582694792293, 3.80312083762382190838890465070, 4.96603081721109267159462794583, 5.20032903992918423734641365957, 6.22698090917270415767630985565, 7.37858735152002197810473691779, 8.119482517484595697113128409814, 8.685014678482404911651142867152

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