Properties

Label 2-4032-48.11-c1-0-38
Degree $2$
Conductor $4032$
Sign $-0.693 + 0.720i$
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  + (−1.41 + 1.41i)5-s + 7-s + (−1.89 − 1.89i)11-s + (0.853 − 0.853i)13-s + 2.59i·17-s + 0.206i·23-s + 0.999i·25-s + (−3.10 − 3.10i)29-s + 1.70i·31-s + (−1.41 + 1.41i)35-s + (−1 − i)37-s − 11.0·41-s + (7.12 − 7.12i)43-s + 3.24·47-s + 49-s + ⋯
L(s)  = 1  + (−0.632 + 0.632i)5-s + 0.377·7-s + (−0.572 − 0.572i)11-s + (0.236 − 0.236i)13-s + 0.628i·17-s + 0.0431i·23-s + 0.199i·25-s + (−0.576 − 0.576i)29-s + 0.306i·31-s + (−0.239 + 0.239i)35-s + (−0.164 − 0.164i)37-s − 1.72·41-s + (1.08 − 1.08i)43-s + 0.472·47-s + 0.142·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.693 + 0.720i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.693 + 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $-0.693 + 0.720i$
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: \((2,\ 4032,\ (\ :1/2),\ -0.693 + 0.720i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3635547844\)
\(L(\frac12)\) \(\approx\) \(0.3635547844\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.41 - 1.41i)T - 5iT^{2} \)
11 \( 1 + (1.89 + 1.89i)T + 11iT^{2} \)
13 \( 1 + (-0.853 + 0.853i)T - 13iT^{2} \)
17 \( 1 - 2.59iT - 17T^{2} \)
19 \( 1 + 19iT^{2} \)
23 \( 1 - 0.206iT - 23T^{2} \)
29 \( 1 + (3.10 + 3.10i)T + 29iT^{2} \)
31 \( 1 - 1.70iT - 31T^{2} \)
37 \( 1 + (1 + i)T + 37iT^{2} \)
41 \( 1 + 11.0T + 41T^{2} \)
43 \( 1 + (-7.12 + 7.12i)T - 43iT^{2} \)
47 \( 1 - 3.24T + 47T^{2} \)
53 \( 1 + (0.722 - 0.722i)T - 53iT^{2} \)
59 \( 1 + (5.00 + 5.00i)T + 59iT^{2} \)
61 \( 1 + (3.14 - 3.14i)T - 61iT^{2} \)
67 \( 1 + (4.14 + 4.14i)T + 67iT^{2} \)
71 \( 1 - 10.0iT - 71T^{2} \)
73 \( 1 - 5.66iT - 73T^{2} \)
79 \( 1 + 3.41iT - 79T^{2} \)
83 \( 1 + (-7.27 + 7.27i)T - 83iT^{2} \)
89 \( 1 + 17.1T + 89T^{2} \)
97 \( 1 + 16.5T + 97T^{2} \)
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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

−8.155670348812207404160186988785, −7.47507295396219633192558899716, −6.83920743837644002351933114089, −5.87268634461628294946316882598, −5.30128305884137017704312984015, −4.21236112033189944624198091600, −3.52280476773060428213073637344, −2.73183755871121786772755613528, −1.58551218791754511876558404843, −0.10846412090085402637824905712, 1.21665449077925040764236874292, 2.30597280907186412651803080779, 3.35277536190981476418637650682, 4.34690941192334445238943164823, 4.83630595611394473446830608206, 5.61121754028885563826577393693, 6.60461936358432782326033746532, 7.43093659267684727063953957166, 7.937650065089197043577717758117, 8.667454105662195791939917123912

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