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Properties

Label 2-3920-140.103-c0-0-1
Degree $2$
Conductor $3920$
Sign $0.251 - 0.967i$
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  + (−0.793 + 0.608i)5-s + (0.866 + 0.5i)9-s + (0.541 − 0.541i)13-s + (−0.478 + 1.78i)17-s + (0.258 − 0.965i)25-s − 1.41i·29-s + (0.517 + 1.93i)37-s + 0.765i·41-s + (−0.991 + 0.130i)45-s + (−0.366 + 1.36i)53-s + (−1.60 − 0.923i)61-s + (−0.0999 + 0.758i)65-s + (1.78 + 0.478i)73-s + (0.499 + 0.866i)81-s + (−0.707 − 1.70i)85-s + ⋯
L(s)  = 1  + (−0.793 + 0.608i)5-s + (0.866 + 0.5i)9-s + (0.541 − 0.541i)13-s + (−0.478 + 1.78i)17-s + (0.258 − 0.965i)25-s − 1.41i·29-s + (0.517 + 1.93i)37-s + 0.765i·41-s + (−0.991 + 0.130i)45-s + (−0.366 + 1.36i)53-s + (−1.60 − 0.923i)61-s + (−0.0999 + 0.758i)65-s + (1.78 + 0.478i)73-s + (0.499 + 0.866i)81-s + (−0.707 − 1.70i)85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.251 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.251 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $0.251 - 0.967i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (3743, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :0),\ 0.251 - 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.103508015\)
\(L(\frac12)\) \(\approx\) \(1.103508015\)
\(L(1)\) 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 \( 1 + (0.793 - 0.608i)T \)
7 \( 1 \)
good3 \( 1 + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.541 + 0.541i)T - iT^{2} \)
17 \( 1 + (0.478 - 1.78i)T + (-0.866 - 0.5i)T^{2} \)
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 \( 1 + (-0.517 - 1.93i)T + (-0.866 + 0.5i)T^{2} \)
41 \( 1 - 0.765iT - T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (-0.866 + 0.5i)T^{2} \)
53 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (1.60 + 0.923i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-1.78 - 0.478i)T + (0.866 + 0.5i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (0.923 - 1.60i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.541 - 0.541i)T + iT^{2} \)
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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

−8.499766548961195795262507515338, −8.015544356940162536396380881383, −7.53163874904741208263837726562, −6.41081883181127234999021853388, −6.19437521664088525944740076618, −4.83908673812844101569307355075, −4.18139737864619545815367028961, −3.49636570101772206873916743144, −2.47596044222976871537252376756, −1.33863972696534547003139124745, 0.68917433027391272190956518837, 1.85631519466179718673316781499, 3.17721926390160841739470778842, 3.96060294550975382071580377397, 4.64584790703359953027521556995, 5.33251073464413249849282197821, 6.43229965401677169920730328138, 7.23038742299550811865789273922, 7.52271290589467752059962885652, 8.714042254812082276051639721817

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