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Properties

Label 2-3332-476.139-c0-0-3
Degree $2$
Conductor $3332$
Sign $0.0290 + 0.999i$
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  + (0.923 − 0.382i)2-s + (0.707 − 0.707i)4-s + (1.63 − 1.08i)5-s + (0.382 − 0.923i)8-s + (−0.923 − 0.382i)9-s + (1.08 − 1.63i)10-s + (−1 + i)13-s − i·16-s + (−0.923 + 0.382i)17-s − 18-s + (0.382 − 1.92i)20-s + (1.08 − 2.63i)25-s + (−0.541 + 1.30i)26-s + (0.324 − 0.216i)29-s + (−0.382 − 0.923i)32-s + ⋯
L(s)  = 1  + (0.923 − 0.382i)2-s + (0.707 − 0.707i)4-s + (1.63 − 1.08i)5-s + (0.382 − 0.923i)8-s + (−0.923 − 0.382i)9-s + (1.08 − 1.63i)10-s + (−1 + i)13-s − i·16-s + (−0.923 + 0.382i)17-s − 18-s + (0.382 − 1.92i)20-s + (1.08 − 2.63i)25-s + (−0.541 + 1.30i)26-s + (0.324 − 0.216i)29-s + (−0.382 − 0.923i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0290 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0290 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.0290 + 0.999i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.0290 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.648019706\)
\(L(\frac12)\) \(\approx\) \(2.648019706\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.923 + 0.382i)T \)
7 \( 1 \)
17 \( 1 + (0.923 - 0.382i)T \)
good3 \( 1 + (0.923 + 0.382i)T^{2} \)
5 \( 1 + (-1.63 + 1.08i)T + (0.382 - 0.923i)T^{2} \)
11 \( 1 + (0.923 - 0.382i)T^{2} \)
13 \( 1 + (1 - i)T - iT^{2} \)
19 \( 1 + (0.707 - 0.707i)T^{2} \)
23 \( 1 + (-0.923 + 0.382i)T^{2} \)
29 \( 1 + (-0.324 + 0.216i)T + (0.382 - 0.923i)T^{2} \)
31 \( 1 + (-0.923 - 0.382i)T^{2} \)
37 \( 1 + (-1.92 - 0.382i)T + (0.923 + 0.382i)T^{2} \)
41 \( 1 + (-0.923 - 0.617i)T + (0.382 + 0.923i)T^{2} \)
43 \( 1 + (-0.707 - 0.707i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \)
59 \( 1 + (-0.707 - 0.707i)T^{2} \)
61 \( 1 + (0.923 - 1.38i)T + (-0.382 - 0.923i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (-0.923 - 0.382i)T^{2} \)
73 \( 1 + (0.324 - 0.216i)T + (0.382 - 0.923i)T^{2} \)
79 \( 1 + (0.923 - 0.382i)T^{2} \)
83 \( 1 + (-0.707 + 0.707i)T^{2} \)
89 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
97 \( 1 + (0.216 + 0.324i)T + (-0.382 + 0.923i)T^{2} \)
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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

−9.004535978926739391162283119462, −7.911793024983258550937901875835, −6.63454176480663788957060539128, −6.19423524645800299174735942103, −5.57071309389841223088579434817, −4.71920183970790751403120097716, −4.28264734424363737392285655803, −2.71885843686669212073336595609, −2.24625084026719771203152254525, −1.19211359868580454245094185473, 2.09009878459122569250476105671, 2.64194241934139287759876031297, 3.19463920623852222311775881511, 4.64462868175144484331631049827, 5.35745360639380433614045149520, 5.96040069747382432645394267519, 6.49379189788093177594779662054, 7.32642431473500000589247113916, 7.952981710545805590920798546625, 9.056760021241794925674171845980

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