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Properties

Label 2-3332-3332.3263-c0-0-1
Degree $2$
Conductor $3332$
Sign $-0.926 - 0.375i$
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  + (0.900 + 0.433i)2-s + (0.433 + 1.90i)3-s + (0.623 + 0.781i)4-s + (−0.433 + 1.90i)6-s + (0.974 − 0.222i)7-s + (0.222 + 0.974i)8-s + (−2.52 + 1.21i)9-s + (0.781 + 0.376i)11-s + (−1.21 + 1.52i)12-s + (−1.62 − 0.781i)13-s + (0.974 + 0.222i)14-s + (−0.222 + 0.974i)16-s + (0.623 − 0.781i)17-s − 2.80·18-s + (0.846 + 1.75i)21-s + (0.541 + 0.678i)22-s + ⋯
L(s)  = 1  + (0.900 + 0.433i)2-s + (0.433 + 1.90i)3-s + (0.623 + 0.781i)4-s + (−0.433 + 1.90i)6-s + (0.974 − 0.222i)7-s + (0.222 + 0.974i)8-s + (−2.52 + 1.21i)9-s + (0.781 + 0.376i)11-s + (−1.21 + 1.52i)12-s + (−1.62 − 0.781i)13-s + (0.974 + 0.222i)14-s + (−0.222 + 0.974i)16-s + (0.623 − 0.781i)17-s − 2.80·18-s + (0.846 + 1.75i)21-s + (0.541 + 0.678i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 - 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 - 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-0.926 - 0.375i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (3263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ -0.926 - 0.375i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.588429190\)
\(L(\frac12)\) \(\approx\) \(2.588429190\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.900 - 0.433i)T \)
7 \( 1 + (-0.974 + 0.222i)T \)
17 \( 1 + (-0.623 + 0.781i)T \)
good3 \( 1 + (-0.433 - 1.90i)T + (-0.900 + 0.433i)T^{2} \)
5 \( 1 + (0.900 - 0.433i)T^{2} \)
11 \( 1 + (-0.781 - 0.376i)T + (0.623 + 0.781i)T^{2} \)
13 \( 1 + (1.62 + 0.781i)T + (0.623 + 0.781i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.541 + 0.678i)T + (-0.222 + 0.974i)T^{2} \)
29 \( 1 + (0.222 + 0.974i)T^{2} \)
31 \( 1 - 1.94T + T^{2} \)
37 \( 1 + (0.222 + 0.974i)T^{2} \)
41 \( 1 + (0.900 - 0.433i)T^{2} \)
43 \( 1 + (0.900 + 0.433i)T^{2} \)
47 \( 1 + (-0.623 - 0.781i)T^{2} \)
53 \( 1 + (0.277 + 0.347i)T + (-0.222 + 0.974i)T^{2} \)
59 \( 1 + (0.900 + 0.433i)T^{2} \)
61 \( 1 + (0.222 + 0.974i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-1.21 - 1.52i)T + (-0.222 + 0.974i)T^{2} \)
73 \( 1 + (-0.623 + 0.781i)T^{2} \)
79 \( 1 - 0.867T + T^{2} \)
83 \( 1 + (-0.623 + 0.781i)T^{2} \)
89 \( 1 + (0.400 - 0.193i)T + (0.623 - 0.781i)T^{2} \)
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

−9.216942959544971842643543627453, −8.105874494272073648341098889915, −7.956049579301469208973203934570, −6.86196347446419068078581255035, −5.64132575359613991663494613238, −5.10883514961152540407019229825, −4.52041493401109337489911837641, −4.00354548623843837463063398171, −2.99783523918738259681014161286, −2.31574799004138625653225051349, 1.15215486302230537003789172583, 1.95040540858018553602810501150, 2.51101119212846296005406007100, 3.59506990121780447685758916552, 4.62648763225245566843811929897, 5.61122042115418984394762448297, 6.27991976741267735100037101896, 6.86660251904756859975224910785, 7.75324839177522967479045258714, 8.081504984212278044529573734169

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