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Properties

Label 2-3960-5.4-c1-0-59
Degree $2$
Conductor $3960$
Sign $0.273 + 0.961i$
Analytic cond. $31.6207$
Root an. cond. $5.62323$
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  + (2.15 − 0.611i)5-s + 4.56i·7-s − 11-s − 7.18i·13-s − 1.73i·17-s − 0.424·19-s − 6.92i·23-s + (4.25 − 2.63i)25-s − 9.38·29-s + 4.56·31-s + (2.79 + 9.81i)35-s − 8.44i·37-s − 7.27·41-s + 4.29i·43-s − 8.86i·47-s + ⋯
L(s)  = 1  + (0.961 − 0.273i)5-s + 1.72i·7-s − 0.301·11-s − 1.99i·13-s − 0.421i·17-s − 0.0974·19-s − 1.44i·23-s + (0.850 − 0.526i)25-s − 1.74·29-s + 0.820·31-s + (0.471 + 1.65i)35-s − 1.38i·37-s − 1.13·41-s + 0.655i·43-s − 1.29i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.273 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.273 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3960\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $0.273 + 0.961i$
Analytic conductor: \(31.6207\)
Root analytic conductor: \(5.62323\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3960} (3169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3960,\ (\ :1/2),\ 0.273 + 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.756144395\)
\(L(\frac12)\) \(\approx\) \(1.756144395\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.15 + 0.611i)T \)
11 \( 1 + T \)
good7 \( 1 - 4.56iT - 7T^{2} \)
13 \( 1 + 7.18iT - 13T^{2} \)
17 \( 1 + 1.73iT - 17T^{2} \)
19 \( 1 + 0.424T + 19T^{2} \)
23 \( 1 + 6.92iT - 23T^{2} \)
29 \( 1 + 9.38T + 29T^{2} \)
31 \( 1 - 4.56T + 31T^{2} \)
37 \( 1 + 8.44iT - 37T^{2} \)
41 \( 1 + 7.27T + 41T^{2} \)
43 \( 1 - 4.29iT - 43T^{2} \)
47 \( 1 + 8.86iT - 47T^{2} \)
53 \( 1 + 7.56iT - 53T^{2} \)
59 \( 1 + 2.11T + 59T^{2} \)
61 \( 1 - 7.38T + 61T^{2} \)
67 \( 1 + 1.07iT - 67T^{2} \)
71 \( 1 - 11.6T + 71T^{2} \)
73 \( 1 + 4.24iT - 73T^{2} \)
79 \( 1 + 14.2T + 79T^{2} \)
83 \( 1 + 3.60iT - 83T^{2} \)
89 \( 1 - 15.4T + 89T^{2} \)
97 \( 1 - 13.4iT - 97T^{2} \)
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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

−8.458374663147965067981416202385, −7.77732515372615359777656648920, −6.63089065494102546824268110447, −5.89177835779962725559559887439, −5.36834796133340039965295216123, −4.95328495937688008842981909145, −3.42741397535891477668371356930, −2.56055107603029508951596236573, −2.05730913077331378672481677909, −0.48584484589124073123146085357, 1.32265604853454116393569699252, 1.92811011974865817444394655695, 3.26862016175723516968527764355, 4.05409960575948907510767820467, 4.73542904271877410321335954286, 5.72714526580551345664617813383, 6.58478441997211288645007319321, 7.04371859482451394021255654876, 7.67824392567798220354777328782, 8.684673121387081755634019999080

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