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Properties

Label 2-3808-952.237-c0-0-4
Degree $2$
Conductor $3808$
Sign $0.309 - 0.951i$
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  + 1.90i·3-s − 1.17i·5-s + 7-s − 2.61·9-s + 2.23·15-s + 17-s + 1.90i·21-s − 0.381·25-s − 3.07i·27-s + 1.61·31-s − 1.17i·35-s − 0.618·41-s + 1.90i·43-s + 3.07i·45-s + 49-s + ⋯
L(s)  = 1  + 1.90i·3-s − 1.17i·5-s + 7-s − 2.61·9-s + 2.23·15-s + 17-s + 1.90i·21-s − 0.381·25-s − 3.07i·27-s + 1.61·31-s − 1.17i·35-s − 0.618·41-s + 1.90i·43-s + 3.07i·45-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3808\)    =    \(2^{5} \cdot 7 \cdot 17\)
Sign: $0.309 - 0.951i$
Analytic conductor: \(1.90043\)
Root analytic conductor: \(1.37856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3808} (3569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3808,\ (\ :0),\ 0.309 - 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.424891699\)
\(L(\frac12)\) \(\approx\) \(1.424891699\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - T \)
17 \( 1 - T \)
good3 \( 1 - 1.90iT - T^{2} \)
5 \( 1 + 1.17iT - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - 1.61T + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + 0.618T + T^{2} \)
43 \( 1 - 1.90iT - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 1.17iT - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + 1.90iT - T^{2} \)
67 \( 1 - 1.17iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 0.618T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - 1.61T + T^{2} \)
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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

−8.901873069103863427741912746592, −8.264139518971691248545644220403, −7.83373763752806333799851058333, −6.26134955352158490656437796660, −5.44680303014830919589535263719, −4.84430981138075124449622775563, −4.52203560917423071567810775307, −3.64511799659171553158835914152, −2.68638259072815263849897155521, −1.16836248226138173994248206264, 1.01955770609237974815582454792, 2.01237503475484045843156791844, 2.69590975403555762289568677486, 3.57189688292854495768567703790, 5.03216854716428668642060173521, 5.80113768708420282360123574107, 6.53873918865929285003444528177, 7.09562211211563206774181965664, 7.68034663816381061871127267557, 8.220008707161007166195492136074

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