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Properties

Label 2-4032-48.11-c1-0-15
Degree $2$
Conductor $4032$
Sign $-0.375 - 0.926i$
Analytic cond. $32.1956$
Root an. cond. $5.67412$
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  + (−2.01 + 2.01i)5-s + 7-s + (3.69 + 3.69i)11-s + (2.62 − 2.62i)13-s + 6.01i·17-s + (−1.07 − 1.07i)19-s + 3.99i·23-s − 3.11i·25-s + (−2.70 − 2.70i)29-s − 1.17i·31-s + (−2.01 + 2.01i)35-s + (3.35 + 3.35i)37-s + 8.80·41-s + (−7.99 + 7.99i)43-s + 3.66·47-s + ⋯
L(s)  = 1  + (−0.900 + 0.900i)5-s + 0.377·7-s + (1.11 + 1.11i)11-s + (0.727 − 0.727i)13-s + 1.45i·17-s + (−0.246 − 0.246i)19-s + 0.833i·23-s − 0.622i·25-s + (−0.501 − 0.501i)29-s − 0.211i·31-s + (−0.340 + 0.340i)35-s + (0.552 + 0.552i)37-s + 1.37·41-s + (−1.21 + 1.21i)43-s + 0.534·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.375 - 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.375 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $-0.375 - 0.926i$
Analytic conductor: \(32.1956\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4032} (3599, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4032,\ (\ :1/2),\ -0.375 - 0.926i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.600759607\)
\(L(\frac12)\) \(\approx\) \(1.600759607\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 - T \)
good5 \( 1 + (2.01 - 2.01i)T - 5iT^{2} \)
11 \( 1 + (-3.69 - 3.69i)T + 11iT^{2} \)
13 \( 1 + (-2.62 + 2.62i)T - 13iT^{2} \)
17 \( 1 - 6.01iT - 17T^{2} \)
19 \( 1 + (1.07 + 1.07i)T + 19iT^{2} \)
23 \( 1 - 3.99iT - 23T^{2} \)
29 \( 1 + (2.70 + 2.70i)T + 29iT^{2} \)
31 \( 1 + 1.17iT - 31T^{2} \)
37 \( 1 + (-3.35 - 3.35i)T + 37iT^{2} \)
41 \( 1 - 8.80T + 41T^{2} \)
43 \( 1 + (7.99 - 7.99i)T - 43iT^{2} \)
47 \( 1 - 3.66T + 47T^{2} \)
53 \( 1 + (-9.44 + 9.44i)T - 53iT^{2} \)
59 \( 1 + (3.29 + 3.29i)T + 59iT^{2} \)
61 \( 1 + (-5.05 + 5.05i)T - 61iT^{2} \)
67 \( 1 + (2.32 + 2.32i)T + 67iT^{2} \)
71 \( 1 - 2.79iT - 71T^{2} \)
73 \( 1 + 4.79iT - 73T^{2} \)
79 \( 1 - 1.49iT - 79T^{2} \)
83 \( 1 + (5.86 - 5.86i)T - 83iT^{2} \)
89 \( 1 - 7.45T + 89T^{2} \)
97 \( 1 + 5.33T + 97T^{2} \)
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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

−8.470412231472303518121624108395, −7.889667300090671771916626202874, −7.28263612714406372556019895228, −6.50442276484413332332125383986, −5.91405536539342339150682315776, −4.75028305905752882906599180435, −3.87987368593960298628191975335, −3.53839355142752820557346308442, −2.26697601423948294604579067973, −1.24812707280513481790292995281, 0.53542806787627117046453880538, 1.34790406238180729638529943586, 2.72420093874030622569414685587, 3.93180050560986075725631356195, 4.17189589455646062046411333742, 5.16934804645909645364336700308, 5.96099860854940535799892023219, 6.82053285376124740511443131634, 7.53710248581309252703022954884, 8.380801132120539078050948687934

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