↑

Properties

Label 2-3332-3332.2447-c0-0-3
Degree $2$
Conductor $3332$
Sign $0.481 + 0.876i$
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  + (0.365 − 0.930i)2-s + (−0.0111 + 0.149i)3-s + (−0.733 − 0.680i)4-s + (0.134 + 0.0648i)6-s + (0.900 − 0.433i)7-s + (−0.900 + 0.433i)8-s + (0.966 + 0.145i)9-s + (0.722 − 0.108i)11-s + (0.109 − 0.101i)12-s + (0.455 + 0.571i)13-s + (−0.0747 − 0.997i)14-s + (0.0747 + 0.997i)16-s + (0.955 + 0.294i)17-s + (0.488 − 0.846i)18-s + (0.0546 + 0.139i)21-s + (0.162 − 0.712i)22-s + ⋯
L(s)  = 1  + (0.365 − 0.930i)2-s + (−0.0111 + 0.149i)3-s + (−0.733 − 0.680i)4-s + (0.134 + 0.0648i)6-s + (0.900 − 0.433i)7-s + (−0.900 + 0.433i)8-s + (0.966 + 0.145i)9-s + (0.722 − 0.108i)11-s + (0.109 − 0.101i)12-s + (0.455 + 0.571i)13-s + (−0.0747 − 0.997i)14-s + (0.0747 + 0.997i)16-s + (0.955 + 0.294i)17-s + (0.488 − 0.846i)18-s + (0.0546 + 0.139i)21-s + (0.162 − 0.712i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.481 + 0.876i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (2447, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.481 + 0.876i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.757025568\)
\(L(\frac12)\) \(\approx\) \(1.757025568\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.365 + 0.930i)T \)
7 \( 1 + (-0.900 + 0.433i)T \)
17 \( 1 + (-0.955 - 0.294i)T \)
good3 \( 1 + (0.0111 - 0.149i)T + (-0.988 - 0.149i)T^{2} \)
5 \( 1 + (-0.365 - 0.930i)T^{2} \)
11 \( 1 + (-0.722 + 0.108i)T + (0.955 - 0.294i)T^{2} \)
13 \( 1 + (-0.455 - 0.571i)T + (-0.222 + 0.974i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (1.19 - 0.367i)T + (0.826 - 0.563i)T^{2} \)
29 \( 1 + (0.900 - 0.433i)T^{2} \)
31 \( 1 + (0.900 - 1.56i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.0747 + 0.997i)T^{2} \)
41 \( 1 + (-0.623 + 0.781i)T^{2} \)
43 \( 1 + (-0.623 - 0.781i)T^{2} \)
47 \( 1 + (0.733 + 0.680i)T^{2} \)
53 \( 1 + (1.21 + 1.12i)T + (0.0747 + 0.997i)T^{2} \)
59 \( 1 + (-0.365 + 0.930i)T^{2} \)
61 \( 1 + (-0.0747 + 0.997i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.367 + 1.61i)T + (-0.900 - 0.433i)T^{2} \)
73 \( 1 + (0.733 - 0.680i)T^{2} \)
79 \( 1 + (0.988 + 1.71i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.222 + 0.974i)T^{2} \)
89 \( 1 + (0.147 + 0.0222i)T + (0.955 + 0.294i)T^{2} \)
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

−8.876518667982110483115303704246, −8.041821987511253315095597841089, −7.22133719107230246322451679055, −6.31933196898849435699427717901, −5.35571524392628963591512710234, −4.68437890968365946780229260621, −3.88316028410984038302803113404, −3.38298266180389679943794909545, −1.68784286700694512684485983916, −1.46237620828986753753954683686, 1.20428093689085394173066419885, 2.52562505145583689150304974782, 3.84472277537081628355889239674, 4.29804224163850949562555438930, 5.26510354460159993594255532510, 5.95653514296305028677058144967, 6.62030809972976821056019763691, 7.58739978104213017043647792654, 7.938446414148832100804882073013, 8.699648136904427596636333757796

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