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Properties

Label 2-3332-68.59-c0-0-1
Degree $2$
Conductor $3332$
Sign $0.997 - 0.0758i$
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  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (0.707 + 0.292i)5-s + (0.707 + 0.707i)8-s + (−0.707 − 0.707i)9-s + (−0.707 + 0.292i)10-s − 1.00·16-s + (0.707 − 0.707i)17-s + 1.00·18-s + (0.292 − 0.707i)20-s + (−0.292 − 0.292i)25-s + (0.707 + 0.292i)29-s + (0.707 − 0.707i)32-s + 1.00i·34-s + (−0.707 + 0.707i)36-s + (0.292 − 0.707i)37-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (0.707 + 0.292i)5-s + (0.707 + 0.707i)8-s + (−0.707 − 0.707i)9-s + (−0.707 + 0.292i)10-s − 1.00·16-s + (0.707 − 0.707i)17-s + 1.00·18-s + (0.292 − 0.707i)20-s + (−0.292 − 0.292i)25-s + (0.707 + 0.292i)29-s + (0.707 − 0.707i)32-s + 1.00i·34-s + (−0.707 + 0.707i)36-s + (0.292 − 0.707i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.997 - 0.0758i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (1079, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.997 - 0.0758i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9421070093\)
\(L(\frac12)\) \(\approx\) \(0.9421070093\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 \)
17 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (0.707 + 0.707i)T^{2} \)
5 \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \)
11 \( 1 + (0.707 - 0.707i)T^{2} \)
13 \( 1 - T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 + (0.707 - 0.707i)T^{2} \)
29 \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \)
31 \( 1 + (0.707 + 0.707i)T^{2} \)
37 \( 1 + (-0.292 + 0.707i)T + (-0.707 - 0.707i)T^{2} \)
41 \( 1 + (-1.70 + 0.707i)T + (0.707 - 0.707i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (1 - i)T - iT^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 + (-1.70 + 0.707i)T + (0.707 - 0.707i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.707 + 0.707i)T^{2} \)
73 \( 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2} \)
79 \( 1 + (0.707 - 0.707i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + 2iT - T^{2} \)
97 \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \)
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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

−8.929050911812532295571743181642, −8.048431797944206073294240653001, −7.35725301672674617264393680956, −6.54889509110673790873987493931, −5.92382771393812870142007529452, −5.40727290182896558775355125325, −4.34711649536263176711536211360, −3.07206091283863621809561178184, −2.16111172548121158707996751811, −0.821717434010725781486152411641, 1.19184740120769111915557366517, 2.18186718006145966775075437264, 2.95020549410106364208670631832, 3.97289982351400572450958247198, 4.95418161849566728379002367075, 5.77978260182117092063982908069, 6.58310090312581522576963180669, 7.70807169230564565530702917315, 8.136176203747341750846830767904, 8.856135268881416252023516474432

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