↑

Properties

Label 2-3332-476.75-c0-0-3
Degree $2$
Conductor $3332$
Sign $-0.637 - 0.770i$
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  + (−0.608 − 0.793i)2-s + (−0.258 + 0.965i)4-s + (0.357 − 1.05i)5-s + (0.923 − 0.382i)8-s + (−0.991 − 0.130i)9-s + (−1.05 + 0.357i)10-s + (−1 − i)13-s + (−0.866 − 0.499i)16-s + (−0.991 + 0.130i)17-s + (0.499 + 0.866i)18-s + (0.923 + 0.617i)20-s + (−0.186 − 0.142i)25-s + (−0.184 + 1.40i)26-s + (−1.63 + 0.324i)29-s + (0.130 + 0.991i)32-s + ⋯
L(s)  = 1  + (−0.608 − 0.793i)2-s + (−0.258 + 0.965i)4-s + (0.357 − 1.05i)5-s + (0.923 − 0.382i)8-s + (−0.991 − 0.130i)9-s + (−1.05 + 0.357i)10-s + (−1 − i)13-s + (−0.866 − 0.499i)16-s + (−0.991 + 0.130i)17-s + (0.499 + 0.866i)18-s + (0.923 + 0.617i)20-s + (−0.186 − 0.142i)25-s + (−0.184 + 1.40i)26-s + (−1.63 + 0.324i)29-s + (0.130 + 0.991i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-0.637 - 0.770i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (1979, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ -0.637 - 0.770i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2201727147\)
\(L(\frac12)\) \(\approx\) \(0.2201727147\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.608 + 0.793i)T \)
7 \( 1 \)
17 \( 1 + (0.991 - 0.130i)T \)
good3 \( 1 + (0.991 + 0.130i)T^{2} \)
5 \( 1 + (-0.357 + 1.05i)T + (-0.793 - 0.608i)T^{2} \)
11 \( 1 + (-0.608 - 0.793i)T^{2} \)
13 \( 1 + (1 + i)T + iT^{2} \)
19 \( 1 + (-0.258 + 0.965i)T^{2} \)
23 \( 1 + (-0.991 + 0.130i)T^{2} \)
29 \( 1 + (1.63 - 0.324i)T + (0.923 - 0.382i)T^{2} \)
31 \( 1 + (-0.991 - 0.130i)T^{2} \)
37 \( 1 + (-0.491 - 0.996i)T + (-0.608 + 0.793i)T^{2} \)
41 \( 1 + (0.382 + 0.0761i)T + (0.923 + 0.382i)T^{2} \)
43 \( 1 + (0.707 - 0.707i)T^{2} \)
47 \( 1 + (-0.866 + 0.5i)T^{2} \)
53 \( 1 + (1.83 - 0.241i)T + (0.965 - 0.258i)T^{2} \)
59 \( 1 + (0.258 + 0.965i)T^{2} \)
61 \( 1 + (-1.47 - 1.29i)T + (0.130 + 0.991i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.382 + 0.923i)T^{2} \)
73 \( 1 + (1.09 + 1.25i)T + (-0.130 + 0.991i)T^{2} \)
79 \( 1 + (0.991 - 0.130i)T^{2} \)
83 \( 1 + (0.707 + 0.707i)T^{2} \)
89 \( 1 + (0.517 + 1.93i)T + (-0.866 + 0.5i)T^{2} \)
97 \( 1 + (0.324 + 1.63i)T + (-0.923 + 0.382i)T^{2} \)
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

−8.541403258882402718043446554324, −7.891809248148543511084237580565, −7.08526605957433470498689699333, −5.92840141031400995817855523953, −5.13654836443442186499473789685, −4.47263549332804715036670535589, −3.32408705703150201900701239948, −2.53613925868310853181012624126, −1.52466957322934558575355552672, −0.15009820550092238405954904493, 1.98755029130177010358885936320, 2.59678719798056653351466643487, 3.95244145307370466191685129262, 4.96872804647057930220157152315, 5.71042765411795723594646184143, 6.53818107195583013651792705188, 6.93959309806080574745644897060, 7.72753835260884023600803184433, 8.484294807332236808007268412394, 9.489554412367016842200346676236

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