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Properties

Label 2-3332-476.299-c0-0-1
Degree $2$
Conductor $3332$
Sign $0.893 + 0.448i$
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  + (−0.608 − 0.793i)2-s + (−0.258 + 0.965i)4-s + (1.05 + 0.357i)5-s + (0.923 − 0.382i)8-s + (0.991 + 0.130i)9-s + (−0.357 − 1.05i)10-s + (1 + i)13-s + (−0.866 − 0.499i)16-s + (−0.130 − 0.991i)17-s + (−0.499 − 0.866i)18-s + (−0.617 + 0.923i)20-s + (0.186 + 0.142i)25-s + (0.184 − 1.40i)26-s + (−0.216 − 1.08i)29-s + (0.130 + 0.991i)32-s + ⋯
L(s)  = 1  + (−0.608 − 0.793i)2-s + (−0.258 + 0.965i)4-s + (1.05 + 0.357i)5-s + (0.923 − 0.382i)8-s + (0.991 + 0.130i)9-s + (−0.357 − 1.05i)10-s + (1 + i)13-s + (−0.866 − 0.499i)16-s + (−0.130 − 0.991i)17-s + (−0.499 − 0.866i)18-s + (−0.617 + 0.923i)20-s + (0.186 + 0.142i)25-s + (0.184 − 1.40i)26-s + (−0.216 − 1.08i)29-s + (0.130 + 0.991i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 + 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 + 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.893 + 0.448i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (3155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.893 + 0.448i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.260490341\)
\(L(\frac12)\) \(\approx\) \(1.260490341\)
\(L(1)\) 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 \( 1 + (0.130 + 0.991i)T \)
good3 \( 1 + (-0.991 - 0.130i)T^{2} \)
5 \( 1 + (-1.05 - 0.357i)T + (0.793 + 0.608i)T^{2} \)
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 \( 1 + (0.216 + 1.08i)T + (-0.923 + 0.382i)T^{2} \)
31 \( 1 + (0.991 + 0.130i)T^{2} \)
37 \( 1 + (-1.49 + 0.735i)T + (0.608 - 0.793i)T^{2} \)
41 \( 1 + (0.0761 - 0.382i)T + (-0.923 - 0.382i)T^{2} \)
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 \( 1 + (1.29 - 1.47i)T + (-0.130 - 0.991i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.382 - 0.923i)T^{2} \)
73 \( 1 + (1.25 - 1.09i)T + (0.130 - 0.991i)T^{2} \)
79 \( 1 + (-0.991 + 0.130i)T^{2} \)
83 \( 1 + (0.707 + 0.707i)T^{2} \)
89 \( 1 + (-0.866 + 0.5i)T^{2} \)
97 \( 1 + (-1.63 + 0.324i)T + (0.923 - 0.382i)T^{2} \)
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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

−9.220277042375867753857532785913, −8.076528755421906125418161609008, −7.36920567138313603483294043605, −6.59635666399401870136614512860, −5.89684590354402991110130771020, −4.60525947405653588351217057143, −4.06412250101464644010068383183, −2.89000326733841961606683478413, −2.06778309368915818642433088642, −1.26242599526066894770155284304, 1.20835503897156078869665729075, 1.83811639308141362873532887781, 3.38809986507536881825822977876, 4.51876462746256053539642016659, 5.25267308233358990546222721439, 6.19442592356842810173246289385, 6.33038074695391050446788874403, 7.52391157952636980027985164993, 8.065966568746956959050157990284, 8.925911487940695155658090908969

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