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Properties

Label 2-4680-5.4-c1-0-54
Degree $2$
Conductor $4680$
Sign $0.987 - 0.158i$
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  + (2.20 − 0.353i)5-s + 1.65i·7-s + 2.94·11-s − i·13-s + 1.46i·17-s − 0.532i·23-s + (4.74 − 1.56i)25-s + 5.70·29-s + (0.585 + 3.65i)35-s − 8.77i·37-s − 1.23·41-s − 1.70i·43-s + 2.70i·47-s + 4.25·49-s + 8.77i·53-s + ⋯
L(s)  = 1  + (0.987 − 0.158i)5-s + 0.625i·7-s + 0.888·11-s − 0.277i·13-s + 0.355i·17-s − 0.111i·23-s + (0.949 − 0.312i)25-s + 1.06·29-s + (0.0989 + 0.617i)35-s − 1.44i·37-s − 0.193·41-s − 0.260i·43-s + 0.394i·47-s + 0.608·49-s + 1.20i·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.158i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4680\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.987 - 0.158i$
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: \((2,\ 4680,\ (\ :1/2),\ 0.987 - 0.158i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.648126596\)
\(L(\frac12)\) \(\approx\) \(2.648126596\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.20 + 0.353i)T \)
13 \( 1 + iT \)
good7 \( 1 - 1.65iT - 7T^{2} \)
11 \( 1 - 2.94T + 11T^{2} \)
17 \( 1 - 1.46iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 0.532iT - 23T^{2} \)
29 \( 1 - 5.70T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 8.77iT - 37T^{2} \)
41 \( 1 + 1.23T + 41T^{2} \)
43 \( 1 + 1.70iT - 43T^{2} \)
47 \( 1 - 2.70iT - 47T^{2} \)
53 \( 1 - 8.77iT - 53T^{2} \)
59 \( 1 + 3.83T + 59T^{2} \)
61 \( 1 + 0.241T + 61T^{2} \)
67 \( 1 + 2.58iT - 67T^{2} \)
71 \( 1 - 2.55T + 71T^{2} \)
73 \( 1 - 0.188iT - 73T^{2} \)
79 \( 1 - 11.0T + 79T^{2} \)
83 \( 1 - 7.91iT - 83T^{2} \)
89 \( 1 - 15.9T + 89T^{2} \)
97 \( 1 + 16.8iT - 97T^{2} \)
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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

−8.496799465786426347216678813969, −7.62067797065872935077353098203, −6.66867562971390741102083084350, −6.13301564415279927914005732505, −5.50180073112236378860051855483, −4.71022832893870307432706222537, −3.78880324415242888171106329853, −2.74987326316847345178145792211, −1.98297838439118852706239349516, −0.972014080780196275614306854551, 0.930820802932312107105393889796, 1.81595419843381136120544762052, 2.84451677172992217417910199597, 3.72280101412368113988303317343, 4.64039212084553207863939246504, 5.29967185489225074446328867681, 6.42198962538044721562265012779, 6.57712641687993784153829490245, 7.45835996173841007320495234491, 8.368170153469564144442113206862

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