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Properties

Label 2-3920-140.39-c0-0-8
Degree $2$
Conductor $3920$
Sign $-0.991 + 0.126i$
Analytic cond. $1.95633$
Root an. cond. $1.39869$
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  + (0.866 − 1.5i)3-s + (−0.5 − 0.866i)5-s + (−1 − 1.73i)9-s − 1.73·15-s + (−0.866 − 1.5i)23-s + (−0.499 + 0.866i)25-s − 1.73·27-s − 29-s − 41-s + 1.73·43-s + (−1 + 1.73i)45-s + (0.5 + 0.866i)61-s + (0.866 − 1.5i)67-s − 3·69-s + (0.866 + 1.49i)75-s + ⋯
L(s)  = 1  + (0.866 − 1.5i)3-s + (−0.5 − 0.866i)5-s + (−1 − 1.73i)9-s − 1.73·15-s + (−0.866 − 1.5i)23-s + (−0.499 + 0.866i)25-s − 1.73·27-s − 29-s − 41-s + 1.73·43-s + (−1 + 1.73i)45-s + (0.5 + 0.866i)61-s + (0.866 − 1.5i)67-s − 3·69-s + (0.866 + 1.49i)75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $-0.991 + 0.126i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (1439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :0),\ -0.991 + 0.126i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.304321408\)
\(L(\frac12)\) \(\approx\) \(1.304321408\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 \)
good3 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 - 1.73T + T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - 1.73T + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.079945827083462721473241028423, −7.80580038519904753069600851201, −6.97394946634529699168248529249, −6.29013004248372156181217495616, −5.42594132980299044061662632055, −4.34794937899349079919028252837, −3.55329299812998712318779805701, −2.49569765649978014150772122331, −1.73455605969730029352639096444, −0.63308463080268752261611707418, 2.07508678600864515345624840860, 2.96499161671025764846899260910, 3.74632592457904606770835707131, 4.06738419002790759168722974944, 5.14491971962024509068373537005, 5.86099364536370215759121436522, 6.96955897054420702917668266681, 7.75405688659345585798528270835, 8.255852435382614280078614128719, 9.161695875616002364993737255196

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