Properties

Label 2-3920-140.3-c0-0-2
Degree $2$
Conductor $3920$
Sign $0.999 + 0.0336i$
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  + (−0.991 + 0.130i)5-s + (−0.866 + 0.5i)9-s + (1.30 − 1.30i)13-s + (−0.739 + 0.198i)17-s + (0.965 − 0.258i)25-s + 1.41i·29-s + (1.93 + 0.517i)37-s − 1.84i·41-s + (0.793 − 0.608i)45-s + (1.36 − 0.366i)53-s + (0.662 − 0.382i)61-s + (−1.12 + 1.46i)65-s + (0.198 + 0.739i)73-s + (0.499 − 0.866i)81-s + (0.707 − 0.292i)85-s + ⋯
L(s)  = 1  + (−0.991 + 0.130i)5-s + (−0.866 + 0.5i)9-s + (1.30 − 1.30i)13-s + (−0.739 + 0.198i)17-s + (0.965 − 0.258i)25-s + 1.41i·29-s + (1.93 + 0.517i)37-s − 1.84i·41-s + (0.793 − 0.608i)45-s + (1.36 − 0.366i)53-s + (0.662 − 0.382i)61-s + (−1.12 + 1.46i)65-s + (0.198 + 0.739i)73-s + (0.499 − 0.866i)81-s + (0.707 − 0.292i)85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0336i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $0.999 + 0.0336i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (2383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :0),\ 0.999 + 0.0336i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9923478402\)
\(L(\frac12)\) \(\approx\) \(0.9923478402\)
\(L(1)\) 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 \( 1 + (0.991 - 0.130i)T \)
7 \( 1 \)
good3 \( 1 + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-1.30 + 1.30i)T - iT^{2} \)
17 \( 1 + (0.739 - 0.198i)T + (0.866 - 0.5i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.866 + 0.5i)T^{2} \)
29 \( 1 - 1.41iT - T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-1.93 - 0.517i)T + (0.866 + 0.5i)T^{2} \)
41 \( 1 + 1.84iT - T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (0.866 + 0.5i)T^{2} \)
53 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.662 + 0.382i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.198 - 0.739i)T + (-0.866 + 0.5i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-1.30 - 1.30i)T + iT^{2} \)
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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

−8.564851719676318682615960673103, −8.012096357538373957010166915185, −7.31444955242770765396265256093, −6.41553970235025093070722201913, −5.65576990759260960505105057192, −4.92631672733976368179367818382, −3.89186790067308571823766271824, −3.28506403691536844211228298338, −2.38192765883766882814382479588, −0.822077296150696401396145358258, 0.869743149276443668558106479813, 2.30324857058388436763044949293, 3.31046664378100697153528092199, 4.13930330983861298642817261116, 4.59242978707423446848287961183, 5.94100884482617054555941185440, 6.33191037995147901494297695319, 7.21051963751599658912589412151, 8.040407937976999108385496979436, 8.660884479677266027421476830902

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