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Properties

Label 2-3808-3808.3093-c0-0-5
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  + (−0.891 − 0.453i)2-s + (1.84 − 0.763i)3-s + (0.587 + 0.809i)4-s + (−0.399 + 0.965i)5-s + (−1.98 − 0.156i)6-s + (0.707 + 0.707i)7-s + (−0.156 − 0.987i)8-s + (2.10 − 2.10i)9-s + (0.794 − 0.678i)10-s + (1.70 + 1.04i)12-s + (−0.309 − 0.951i)14-s + 2.08i·15-s + (−0.309 + 0.951i)16-s − i·17-s + (−2.82 + 0.919i)18-s + ⋯
L(s)  = 1  + (−0.891 − 0.453i)2-s + (1.84 − 0.763i)3-s + (0.587 + 0.809i)4-s + (−0.399 + 0.965i)5-s + (−1.98 − 0.156i)6-s + (0.707 + 0.707i)7-s + (−0.156 − 0.987i)8-s + (2.10 − 2.10i)9-s + (0.794 − 0.678i)10-s + (1.70 + 1.04i)12-s + (−0.309 − 0.951i)14-s + 2.08i·15-s + (−0.309 + 0.951i)16-s − i·17-s + (−2.82 + 0.919i)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: $\chi_{3808} (3093, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3808,\ (\ :0),\ 0.831 + 0.555i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.735182533\)
\(L(\frac12)\) \(\approx\) \(1.735182533\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.891 + 0.453i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
17 \( 1 + iT \)
good3 \( 1 + (-1.84 + 0.763i)T + (0.707 - 0.707i)T^{2} \)
5 \( 1 + (0.399 - 0.965i)T + (-0.707 - 0.707i)T^{2} \)
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 \( 1 + 1.61T + T^{2} \)
37 \( 1 + (0.707 + 0.707i)T^{2} \)
41 \( 1 + (-0.437 + 0.437i)T - iT^{2} \)
43 \( 1 + (-1.40 - 0.581i)T + (0.707 + 0.707i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (0.431 + 0.178i)T + (0.707 + 0.707i)T^{2} \)
59 \( 1 + (0.707 + 0.707i)T^{2} \)
61 \( 1 + (-0.144 + 0.0600i)T + (0.707 - 0.707i)T^{2} \)
67 \( 1 + (1.79 - 0.744i)T + (0.707 - 0.707i)T^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 + (1.26 - 1.26i)T - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (0.707 - 0.707i)T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 + 0.312T + T^{2} \)
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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

−8.656018247697836241172255719718, −7.913369755834157983232969404453, −7.28544362525349604902473821146, −7.11649380365519185913604281375, −5.98477640933943450798636176260, −4.34905309751552634578316664804, −3.46416038552818959516292458513, −2.79252869485261193126758995328, −2.28068133989515754350352580782, −1.33681695998981928058494434061, 1.36665105871206143372463233264, 2.08333300721499710021826087433, 3.30282161001141153782728478825, 4.24670188848403994767315032047, 4.68589039847777592632955905059, 5.69124173671094522658253436976, 7.09938866695727644459531909972, 7.68630105563150734356371123895, 8.096247387074179449215996984122, 8.797203119174070195323187953777

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