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Properties

Label 2-3800-152.35-c0-0-4
Degree $2$
Conductor $3800$
Sign $0.756 - 0.654i$
Analytic cond. $1.89644$
Root an. cond. $1.37711$
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  + (0.766 + 0.642i)2-s + (1.76 − 0.642i)3-s + (0.173 + 0.984i)4-s + (1.76 + 0.642i)6-s + (−0.500 + 0.866i)8-s + (1.93 − 1.62i)9-s + (−0.766 + 1.32i)11-s + (0.939 + 1.62i)12-s + (−0.939 + 0.342i)16-s + (−0.766 − 0.642i)17-s + 2.53·18-s + (0.766 − 0.642i)19-s + (−1.43 + 0.524i)22-s + (−0.326 + 1.85i)24-s + (1.43 − 2.49i)27-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)2-s + (1.76 − 0.642i)3-s + (0.173 + 0.984i)4-s + (1.76 + 0.642i)6-s + (−0.500 + 0.866i)8-s + (1.93 − 1.62i)9-s + (−0.766 + 1.32i)11-s + (0.939 + 1.62i)12-s + (−0.939 + 0.342i)16-s + (−0.766 − 0.642i)17-s + 2.53·18-s + (0.766 − 0.642i)19-s + (−1.43 + 0.524i)22-s + (−0.326 + 1.85i)24-s + (1.43 − 2.49i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $0.756 - 0.654i$
Analytic conductor: \(1.89644\)
Root analytic conductor: \(1.37711\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3800} (1251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3800,\ (\ :0),\ 0.756 - 0.654i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.444898518\)
\(L(\frac12)\) \(\approx\) \(3.444898518\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 - 0.642i)T \)
5 \( 1 \)
19 \( 1 + (-0.766 + 0.642i)T \)
good3 \( 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.766 - 0.642i)T^{2} \)
17 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
23 \( 1 + (0.939 - 0.342i)T^{2} \)
29 \( 1 + (-0.173 + 0.984i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2} \)
43 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
47 \( 1 + (-0.173 + 0.984i)T^{2} \)
53 \( 1 + (0.939 - 0.342i)T^{2} \)
59 \( 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2} \)
61 \( 1 + (0.939 - 0.342i)T^{2} \)
67 \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.939 + 0.342i)T^{2} \)
73 \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \)
79 \( 1 + (-0.766 + 0.642i)T^{2} \)
83 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (1.87 + 0.684i)T + (0.766 + 0.642i)T^{2} \)
97 \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \)
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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

−8.536335493528448860733247459210, −7.79994507604771304407138163382, −7.29782506637038413920829691573, −6.95176513340206908944139673891, −5.90352935693784931784445286000, −4.67145421097925607364092112379, −4.30842202658855752992176899649, −3.05293389421121208712818099326, −2.68580524566222964331829090725, −1.77345951636550670047352189157, 1.50786051014669123445812855798, 2.54126919979046509345881878895, 3.09218346791879366697914252569, 3.79334936878745012956437526042, 4.45093724303031218212184007107, 5.37520621460599890396709574141, 6.16459398717167379971306559774, 7.31857728208881232088348872629, 8.058930693361989386536622887179, 8.700061087682479727104865563801

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