Properties

Label 2-3332-476.227-c0-0-2
Degree $2$
Conductor $3332$
Sign $0.315 - 0.948i$
Analytic cond. $1.66288$
Root an. cond. $1.28952$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.130 + 0.991i)2-s + (−0.965 + 0.258i)4-s + (0.0255 − 0.389i)5-s + (−0.382 − 0.923i)8-s + (0.793 + 0.608i)9-s + (0.389 − 0.0255i)10-s + (−1 − i)13-s + (0.866 − 0.5i)16-s + (0.608 + 0.793i)17-s + (−0.499 + 0.866i)18-s + (0.0761 + 0.382i)20-s + (0.840 + 0.110i)25-s + (0.860 − 1.12i)26-s + (−0.324 − 0.216i)29-s + (0.608 + 0.793i)32-s + ⋯
L(s)  = 1  + (0.130 + 0.991i)2-s + (−0.965 + 0.258i)4-s + (0.0255 − 0.389i)5-s + (−0.382 − 0.923i)8-s + (0.793 + 0.608i)9-s + (0.389 − 0.0255i)10-s + (−1 − i)13-s + (0.866 − 0.5i)16-s + (0.608 + 0.793i)17-s + (−0.499 + 0.866i)18-s + (0.0761 + 0.382i)20-s + (0.840 + 0.110i)25-s + (0.860 − 1.12i)26-s + (−0.324 − 0.216i)29-s + (0.608 + 0.793i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.315 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.315 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.315 - 0.948i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.315 - 0.948i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.260830353\)
\(L(\frac12)\) \(\approx\) \(1.260830353\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.130 - 0.991i)T \)
7 \( 1 \)
17 \( 1 + (-0.608 - 0.793i)T \)
good3 \( 1 + (-0.793 - 0.608i)T^{2} \)
5 \( 1 + (-0.0255 + 0.389i)T + (-0.991 - 0.130i)T^{2} \)
11 \( 1 + (-0.130 - 0.991i)T^{2} \)
13 \( 1 + (1 + i)T + iT^{2} \)
19 \( 1 + (-0.965 + 0.258i)T^{2} \)
23 \( 1 + (0.793 - 0.608i)T^{2} \)
29 \( 1 + (0.324 + 0.216i)T + (0.382 + 0.923i)T^{2} \)
31 \( 1 + (0.793 + 0.608i)T^{2} \)
37 \( 1 + (-1.29 - 1.47i)T + (-0.130 + 0.991i)T^{2} \)
41 \( 1 + (-1.38 + 0.923i)T + (0.382 - 0.923i)T^{2} \)
43 \( 1 + (-0.707 + 0.707i)T^{2} \)
47 \( 1 + (0.866 + 0.5i)T^{2} \)
53 \( 1 + (-0.607 + 0.465i)T + (0.258 - 0.965i)T^{2} \)
59 \( 1 + (0.965 + 0.258i)T^{2} \)
61 \( 1 + (0.491 - 0.996i)T + (-0.608 - 0.793i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.923 + 0.382i)T^{2} \)
73 \( 1 + (-1.75 + 0.867i)T + (0.608 - 0.793i)T^{2} \)
79 \( 1 + (-0.793 + 0.608i)T^{2} \)
83 \( 1 + (-0.707 - 0.707i)T^{2} \)
89 \( 1 + (0.866 + 0.5i)T^{2} \)
97 \( 1 + (-1.08 + 1.63i)T + (-0.382 - 0.923i)T^{2} \)
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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

−8.781006203448787612187351015419, −7.87653896867605713838268584145, −7.67606139161472491876847317465, −6.77672675544606691710693578942, −5.89534770204695788715587133382, −5.17157349348225090287874028238, −4.59806758497276383273185319618, −3.72470528325408312983827773411, −2.58592458138590962468269932353, −1.06352760584071827954714501542, 0.977053880368121569984014656510, 2.19459518481256367512294349062, 2.95127247097228980962300889175, 3.97184813022316611645190348649, 4.56234685467900187890377288372, 5.42347637440305190712171010385, 6.42968889500937148686319057707, 7.21168515630980874764259205594, 7.893031555386997958214225059611, 9.119075198068544859131457406102

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