↑

Properties

Label 2-3332-476.67-c0-0-3
Degree $2$
Conductor $3332$
Sign $-0.991 + 0.126i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s − 0.999·8-s + (0.5 + 0.866i)9-s − 2·13-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (−0.499 + 0.866i)18-s + (−0.5 + 0.866i)25-s + (−1 − 1.73i)26-s + (0.499 − 0.866i)32-s − 0.999·34-s − 0.999·36-s − 0.999·50-s + (0.999 − 1.73i)52-s + (−1 + 1.73i)53-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s − 0.999·8-s + (0.5 + 0.866i)9-s − 2·13-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (−0.499 + 0.866i)18-s + (−0.5 + 0.866i)25-s + (−1 − 1.73i)26-s + (0.499 − 0.866i)32-s − 0.999·34-s − 0.999·36-s − 0.999·50-s + (0.999 − 1.73i)52-s + (−1 + 1.73i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-0.991 + 0.126i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ -0.991 + 0.126i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9486639921\)
\(L(\frac12)\) \(\approx\) \(0.9486639921\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 \)
17 \( 1 + (0.5 - 0.866i)T \)
good3 \( 1 + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + 2T + T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.118221012211938747699426161632, −8.147287188195646889761782045648, −7.55801255485593888238791729295, −7.10413381048564031751491802937, −6.19367765564667667915382483722, −5.31762033894491866688723147196, −4.72767493214349438419493053847, −4.05679493636604253725648459282, −2.89528181419206752831906828286, −1.97053740060680400843865440243, 0.44385434784124211530261702712, 1.95152508359678581101723423990, 2.72259065586524609983632754041, 3.65039299881290295411485060490, 4.61226772355878742124274408018, 5.00527919875885398536363600171, 6.10221722846231853893072596094, 6.83823250939125812839740791210, 7.57318488192877308073674614286, 8.666240368694767713621973908942

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