Properties

Label 2-3920-140.47-c0-0-2
Degree $2$
Conductor $3920$
Sign $0.574 + 0.818i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $0.574 + 0.818i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :0),\ 0.574 + 0.818i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.329987573\)
\(L(\frac12)\) \(\approx\) \(1.329987573\)
\(L(1)\) 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

−8.543017547024085620746081597387, −7.78642008893395940878546621387, −7.13131491483062938317060670129, −6.00908453497815629050060946660, −5.74340856966119873623902851084, −4.99438777957034746616791217312, −3.84254203410227553659008452704, −2.78818017277855577153989438057, −2.33805202336877140262489564712, −0.76041929258738686985282340500, 1.41574069744748303335841413108, 2.42049997351164693120069882949, 3.02842921073400671738883633525, 4.45168176111498081840633512469, 5.02855750825867026041379629763, 5.75526208065868474551141263559, 6.51336881562603744127750521525, 7.24974522090137060471140184941, 8.057956967225917975266430304013, 8.869468433389075205640564751079

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