Properties

Label 2-3120-5.4-c1-0-52
Degree $2$
Conductor $3120$
Sign $0.218 + 0.975i$
Analytic cond. $24.9133$
Root an. cond. $4.99132$
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  i·3-s + (0.489 + 2.18i)5-s − 2.82i·7-s − 9-s + 2.97·11-s i·13-s + (2.18 − 0.489i)15-s − 1.44i·17-s + 4.17·19-s − 2.82·21-s − 6.21i·23-s + (−4.52 + 2.13i)25-s + i·27-s + 0.828·29-s − 1.65·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (0.218 + 0.975i)5-s − 1.06i·7-s − 0.333·9-s + 0.898·11-s − 0.277i·13-s + (0.563 − 0.126i)15-s − 0.350i·17-s + 0.956·19-s − 0.617·21-s − 1.29i·23-s + (−0.904 + 0.427i)25-s + 0.192i·27-s + 0.153·29-s − 0.297·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.218 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.218 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3120\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.218 + 0.975i$
Analytic conductor: \(24.9133\)
Root analytic conductor: \(4.99132\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3120} (1249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3120,\ (\ :1/2),\ 0.218 + 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.871343433\)
\(L(\frac12)\) \(\approx\) \(1.871343433\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
5 \( 1 + (-0.489 - 2.18i)T \)
13 \( 1 + iT \)
good7 \( 1 + 2.82iT - 7T^{2} \)
11 \( 1 - 2.97T + 11T^{2} \)
17 \( 1 + 1.44iT - 17T^{2} \)
19 \( 1 - 4.17T + 19T^{2} \)
23 \( 1 + 6.21iT - 23T^{2} \)
29 \( 1 - 0.828T + 29T^{2} \)
31 \( 1 + 1.65T + 31T^{2} \)
37 \( 1 - 0.0418iT - 37T^{2} \)
41 \( 1 + 4.36T + 41T^{2} \)
43 \( 1 - 8.99iT - 43T^{2} \)
47 \( 1 + 4.40iT - 47T^{2} \)
53 \( 1 + 6.72iT - 53T^{2} \)
59 \( 1 + 2.97T + 59T^{2} \)
61 \( 1 - 6.27T + 61T^{2} \)
67 \( 1 - 0.727iT - 67T^{2} \)
71 \( 1 - 8.32T + 71T^{2} \)
73 \( 1 + 6.87iT - 73T^{2} \)
79 \( 1 - 6.11T + 79T^{2} \)
83 \( 1 + 14.7iT - 83T^{2} \)
89 \( 1 + 14.7T + 89T^{2} \)
97 \( 1 - 6.78iT - 97T^{2} \)
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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

−8.357241010745820808810447418542, −7.62402541354264772108199073164, −6.90071380445065797278955066029, −6.59346357845957220620856396956, −5.65878772265783646882731391954, −4.58492885553191831515932234613, −3.62537500894431727205873854677, −2.91610400432112195044841363918, −1.77258318440589582506060438829, −0.63952080278287888236909550906, 1.20716438974920495500014205516, 2.20509712307117396816185818593, 3.44004203616959253182791763118, 4.16469505820736321944323300096, 5.20948592021318711198612397139, 5.55180939162286319418472887664, 6.40458171331801687233126466668, 7.45617998983446247196026542256, 8.388352301550714560009980966987, 8.967241526785085748151595110856

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