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Properties

Label 2-4680-5.4-c1-0-79
Degree $2$
Conductor $4680$
Sign $-0.535 + 0.844i$
Analytic cond. $37.3699$
Root an. cond. $6.11309$
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  + (1.19 − 1.88i)5-s − 2.82i·7-s − 0.393·11-s + i·13-s − 6.21i·17-s + 7.34·19-s + 1.44i·23-s + (−2.13 − 4.52i)25-s − 0.828·29-s + 1.65·31-s + (−5.34 − 3.38i)35-s − 6.78i·37-s − 3.77·41-s − 2.51i·43-s + 3.00i·47-s + ⋯
L(s)  = 1  + (0.535 − 0.844i)5-s − 1.06i·7-s − 0.118·11-s + 0.277i·13-s − 1.50i·17-s + 1.68·19-s + 0.301i·23-s + (−0.427 − 0.904i)25-s − 0.153·29-s + 0.297·31-s + (−0.903 − 0.572i)35-s − 1.11i·37-s − 0.590·41-s − 0.383i·43-s + 0.438i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.535 + 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4680\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.535 + 0.844i$
Analytic conductor: \(37.3699\)
Root analytic conductor: \(6.11309\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4680} (2809, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4680,\ (\ :1/2),\ -0.535 + 0.844i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.981766972\)
\(L(\frac12)\) \(\approx\) \(1.981766972\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-1.19 + 1.88i)T \)
13 \( 1 - iT \)
good7 \( 1 + 2.82iT - 7T^{2} \)
11 \( 1 + 0.393T + 11T^{2} \)
17 \( 1 + 6.21iT - 17T^{2} \)
19 \( 1 - 7.34T + 19T^{2} \)
23 \( 1 - 1.44iT - 23T^{2} \)
29 \( 1 + 0.828T + 29T^{2} \)
31 \( 1 - 1.65T + 31T^{2} \)
37 \( 1 + 6.78iT - 37T^{2} \)
41 \( 1 + 3.77T + 41T^{2} \)
43 \( 1 + 2.51iT - 43T^{2} \)
47 \( 1 - 3.00iT - 47T^{2} \)
53 \( 1 - 9.55iT - 53T^{2} \)
59 \( 1 - 0.393T + 59T^{2} \)
61 \( 1 - 11.0T + 61T^{2} \)
67 \( 1 + 15.5iT - 67T^{2} \)
71 \( 1 + 6.56T + 71T^{2} \)
73 \( 1 - 13.6iT - 73T^{2} \)
79 \( 1 - 14.9T + 79T^{2} \)
83 \( 1 + 16.4iT - 83T^{2} \)
89 \( 1 + 9.67T + 89T^{2} \)
97 \( 1 + 0.0418iT - 97T^{2} \)
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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

−7.901912467904880855548320727912, −7.38812098379143913298162728387, −6.75284001266125136388565490299, −5.67203451497786317365288405707, −5.13238125039582860482290308281, −4.41199049919123644328750863940, −3.54242544302995366478966564975, −2.54669032992145365773585005425, −1.35374502537088204576202948834, −0.56855818053775008485774394956, 1.39324978909080718289549609678, 2.35687604100054920069785469645, 3.07856760024467876489782154713, 3.85421224494548417587346764058, 5.18842292743985972863550575112, 5.58398326391076668941034385672, 6.37398759670818065153137444947, 6.95529174415478699174804878343, 7.953343217438438086781504062086, 8.446777557862659064824629184003

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