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Properties

Label 2-820-820.647-c1-0-14
Degree $2$
Conductor $820$
Sign $-0.968 + 0.250i$
Analytic cond. $6.54773$
Root an. cond. $2.55885$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.33 + 0.461i)2-s − 2.00·3-s + (1.57 − 1.23i)4-s + (−1.66 + 1.48i)5-s + (2.68 − 0.924i)6-s + 4.57i·7-s + (−1.53 + 2.37i)8-s + 1.02·9-s + (1.54 − 2.75i)10-s + (2.54 + 2.54i)11-s + (−3.15 + 2.47i)12-s + 2.88i·13-s + (−2.11 − 6.12i)14-s + (3.34 − 2.98i)15-s + (0.959 − 3.88i)16-s + 0.355i·17-s + ⋯
L(s)  = 1  + (−0.945 + 0.326i)2-s − 1.15·3-s + (0.787 − 0.616i)4-s + (−0.746 + 0.665i)5-s + (1.09 − 0.377i)6-s + 1.73i·7-s + (−0.543 + 0.839i)8-s + 0.340·9-s + (0.488 − 0.872i)10-s + (0.768 + 0.768i)11-s + (−0.911 + 0.713i)12-s + 0.799i·13-s + (−0.564 − 1.63i)14-s + (0.863 − 0.770i)15-s + (0.239 − 0.970i)16-s + 0.0862i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $-0.968 + 0.250i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (647, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ -0.968 + 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0572060 - 0.449967i\)
\(L(\frac12)\) \(\approx\) \(0.0572060 - 0.449967i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.33 - 0.461i)T \)
5 \( 1 + (1.66 - 1.48i)T \)
41 \( 1 + (-6.25 + 1.36i)T \)
good3 \( 1 + 2.00T + 3T^{2} \)
7 \( 1 - 4.57iT - 7T^{2} \)
11 \( 1 + (-2.54 - 2.54i)T + 11iT^{2} \)
13 \( 1 - 2.88iT - 13T^{2} \)
17 \( 1 - 0.355iT - 17T^{2} \)
19 \( 1 + (-4.98 - 4.98i)T + 19iT^{2} \)
23 \( 1 + (-0.157 - 0.157i)T + 23iT^{2} \)
29 \( 1 + (3.65 + 3.65i)T + 29iT^{2} \)
31 \( 1 - 9.42iT - 31T^{2} \)
37 \( 1 + (-3.67 - 3.67i)T + 37iT^{2} \)
43 \( 1 + (4.13 - 4.13i)T - 43iT^{2} \)
47 \( 1 + 3.92T + 47T^{2} \)
53 \( 1 - 3.52iT - 53T^{2} \)
59 \( 1 + 4.54T + 59T^{2} \)
61 \( 1 - 8.21iT - 61T^{2} \)
67 \( 1 - 7.29T + 67T^{2} \)
71 \( 1 + (8.39 + 8.39i)T + 71iT^{2} \)
73 \( 1 + (6.17 + 6.17i)T + 73iT^{2} \)
79 \( 1 + (-7.01 + 7.01i)T - 79iT^{2} \)
83 \( 1 + (1.29 + 1.29i)T + 83iT^{2} \)
89 \( 1 + (-10.8 - 10.8i)T + 89iT^{2} \)
97 \( 1 + 13.6iT - 97T^{2} \)
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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

−10.74021536428097598270786947196, −9.767479751041447791435909632348, −9.072270726746067067588358601188, −8.142196259544496075244546416540, −7.18259582397718575564022234666, −6.33364187519829019073452074710, −5.80324590149233938385778790973, −4.74085878167811804525876899828, −3.04471932969759423676109375475, −1.67009694313387626767739096212, 0.48717365134816936318800212402, 0.962933658231448892030907479387, 3.31561647963823973629702478322, 4.18417561579694948276067409173, 5.37197337891638704628679400631, 6.52011408542559683010892939326, 7.35878069294941479026000837508, 7.949038808549226367600736459391, 9.049856904468288176336308092902, 9.867086027857012373920480370890

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