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Properties

Label 2-4032-48.11-c1-0-19
Degree $2$
Conductor $4032$
Sign $0.272 - 0.962i$
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  + (−0.270 + 0.270i)5-s + 7-s + (3.03 + 3.03i)11-s + (1.28 − 1.28i)13-s + 5.15i·17-s + (5.55 + 5.55i)19-s − 5.69i·23-s + 4.85i·25-s + (−1.94 − 1.94i)29-s + 0.936i·31-s + (−0.270 + 0.270i)35-s + (−1.52 − 1.52i)37-s − 12.4·41-s + (0.346 − 0.346i)43-s + 9.71·47-s + ⋯
L(s)  = 1  + (−0.120 + 0.120i)5-s + 0.377·7-s + (0.915 + 0.915i)11-s + (0.357 − 0.357i)13-s + 1.25i·17-s + (1.27 + 1.27i)19-s − 1.18i·23-s + 0.970i·25-s + (−0.360 − 0.360i)29-s + 0.168i·31-s + (−0.0456 + 0.0456i)35-s + (−0.251 − 0.251i)37-s − 1.94·41-s + (0.0528 − 0.0528i)43-s + 1.41·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.272 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.272 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $0.272 - 0.962i$
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: \((2,\ 4032,\ (\ :1/2),\ 0.272 - 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.981182327\)
\(L(\frac12)\) \(\approx\) \(1.981182327\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.270 - 0.270i)T - 5iT^{2} \)
11 \( 1 + (-3.03 - 3.03i)T + 11iT^{2} \)
13 \( 1 + (-1.28 + 1.28i)T - 13iT^{2} \)
17 \( 1 - 5.15iT - 17T^{2} \)
19 \( 1 + (-5.55 - 5.55i)T + 19iT^{2} \)
23 \( 1 + 5.69iT - 23T^{2} \)
29 \( 1 + (1.94 + 1.94i)T + 29iT^{2} \)
31 \( 1 - 0.936iT - 31T^{2} \)
37 \( 1 + (1.52 + 1.52i)T + 37iT^{2} \)
41 \( 1 + 12.4T + 41T^{2} \)
43 \( 1 + (-0.346 + 0.346i)T - 43iT^{2} \)
47 \( 1 - 9.71T + 47T^{2} \)
53 \( 1 + (7.69 - 7.69i)T - 53iT^{2} \)
59 \( 1 + (2.03 + 2.03i)T + 59iT^{2} \)
61 \( 1 + (-5.86 + 5.86i)T - 61iT^{2} \)
67 \( 1 + (10.7 + 10.7i)T + 67iT^{2} \)
71 \( 1 - 12.5iT - 71T^{2} \)
73 \( 1 + 7.37iT - 73T^{2} \)
79 \( 1 - 13.0iT - 79T^{2} \)
83 \( 1 + (2.53 - 2.53i)T - 83iT^{2} \)
89 \( 1 - 7.04T + 89T^{2} \)
97 \( 1 - 0.334T + 97T^{2} \)
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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

−8.537531965940589568795086623212, −7.84966356763933698218125573041, −7.20339358307379128084519610976, −6.37120821134807010645679621499, −5.68500070944280912848225661112, −4.81155825414483646139517159117, −3.91754341462580738808879301300, −3.35478419998130131828402605722, −1.97059683817187939592787305628, −1.26158824788382511037710429344, 0.62208671191150394248846859435, 1.60371677166777535970200763055, 2.91839208075649837540440234225, 3.57064459723645454069456204432, 4.58819619183169371368258657057, 5.25827168566576871028606841263, 6.05258042126165857313568737722, 6.97863736381211172249608322703, 7.41619889681047625102390525532, 8.433249848936440340324062966815

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