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Properties

Label 2-3332-3332.2311-c0-0-3
Degree $2$
Conductor $3332$
Sign $-0.518 + 0.855i$
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  + (−0.623 − 0.781i)2-s + (0.781 − 0.376i)3-s + (−0.222 + 0.974i)4-s + (−0.781 − 0.376i)6-s + (0.433 + 0.900i)7-s + (0.900 − 0.433i)8-s + (−0.153 + 0.193i)9-s + (−0.974 − 1.22i)11-s + (0.193 + 0.846i)12-s + (−0.777 − 0.974i)13-s + (0.433 − 0.900i)14-s + (−0.900 − 0.433i)16-s + (−0.222 − 0.974i)17-s + 0.246·18-s + (0.678 + 0.541i)21-s + (−0.347 + 1.52i)22-s + ⋯
L(s)  = 1  + (−0.623 − 0.781i)2-s + (0.781 − 0.376i)3-s + (−0.222 + 0.974i)4-s + (−0.781 − 0.376i)6-s + (0.433 + 0.900i)7-s + (0.900 − 0.433i)8-s + (−0.153 + 0.193i)9-s + (−0.974 − 1.22i)11-s + (0.193 + 0.846i)12-s + (−0.777 − 0.974i)13-s + (0.433 − 0.900i)14-s + (−0.900 − 0.433i)16-s + (−0.222 − 0.974i)17-s + 0.246·18-s + (0.678 + 0.541i)21-s + (−0.347 + 1.52i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.518 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.518 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-0.518 + 0.855i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (2311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ -0.518 + 0.855i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9665401345\)
\(L(\frac12)\) \(\approx\) \(0.9665401345\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.623 + 0.781i)T \)
7 \( 1 + (-0.433 - 0.900i)T \)
17 \( 1 + (0.222 + 0.974i)T \)
good3 \( 1 + (-0.781 + 0.376i)T + (0.623 - 0.781i)T^{2} \)
5 \( 1 + (-0.623 + 0.781i)T^{2} \)
11 \( 1 + (0.974 + 1.22i)T + (-0.222 + 0.974i)T^{2} \)
13 \( 1 + (0.777 + 0.974i)T + (-0.222 + 0.974i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.347 + 1.52i)T + (-0.900 - 0.433i)T^{2} \)
29 \( 1 + (0.900 - 0.433i)T^{2} \)
31 \( 1 - 0.867T + T^{2} \)
37 \( 1 + (0.900 - 0.433i)T^{2} \)
41 \( 1 + (-0.623 + 0.781i)T^{2} \)
43 \( 1 + (-0.623 - 0.781i)T^{2} \)
47 \( 1 + (0.222 - 0.974i)T^{2} \)
53 \( 1 + (-0.400 + 1.75i)T + (-0.900 - 0.433i)T^{2} \)
59 \( 1 + (-0.623 - 0.781i)T^{2} \)
61 \( 1 + (0.900 - 0.433i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.193 - 0.846i)T + (-0.900 - 0.433i)T^{2} \)
73 \( 1 + (0.222 + 0.974i)T^{2} \)
79 \( 1 - 1.56T + T^{2} \)
83 \( 1 + (0.222 + 0.974i)T^{2} \)
89 \( 1 + (-1.12 + 1.40i)T + (-0.222 - 0.974i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.534695502563351376263407314847, −8.117348955830325621109446240163, −7.53136961041212137488452486913, −6.48004497451292412416065189554, −5.26239672245037191529920263071, −4.78892689250448382170137530262, −3.25946094679219589207034179230, −2.61803483441737538556162051507, −2.34580724889632683212878475329, −0.64294837586593912064272987617, 1.47588353428623989378900455585, 2.43281982168762132099449891987, 3.75920997125011265416751390843, 4.57574656488125513710024485746, 5.14190742336784369539032992338, 6.27466572532939786555932827973, 7.15270071102489250466764473056, 7.58332863336376039012643333922, 8.206444283847028365008376390205, 9.132574766166031899573995292391

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