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Properties

Label 2-3332-476.419-c0-0-2
Degree $2$
Conductor $3332$
Sign $0.694 + 0.719i$
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  + (−0.923 − 0.382i)2-s + (0.707 + 0.707i)4-s + (−0.216 + 0.324i)5-s + (−0.382 − 0.923i)8-s + (0.923 − 0.382i)9-s + (0.324 − 0.216i)10-s + (−1 − i)13-s + i·16-s + (0.923 + 0.382i)17-s − 18-s + (−0.382 + 0.0761i)20-s + (0.324 + 0.783i)25-s + (0.541 + 1.30i)26-s + (1.08 − 1.63i)29-s + (0.382 − 0.923i)32-s + ⋯
L(s)  = 1  + (−0.923 − 0.382i)2-s + (0.707 + 0.707i)4-s + (−0.216 + 0.324i)5-s + (−0.382 − 0.923i)8-s + (0.923 − 0.382i)9-s + (0.324 − 0.216i)10-s + (−1 − i)13-s + i·16-s + (0.923 + 0.382i)17-s − 18-s + (−0.382 + 0.0761i)20-s + (0.324 + 0.783i)25-s + (0.541 + 1.30i)26-s + (1.08 − 1.63i)29-s + (0.382 − 0.923i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.694 + 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.694 + 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.694 + 0.719i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (1371, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.694 + 0.719i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8239635632\)
\(L(\frac12)\) \(\approx\) \(0.8239635632\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.923 + 0.382i)T \)
7 \( 1 \)
17 \( 1 + (-0.923 - 0.382i)T \)
good3 \( 1 + (-0.923 + 0.382i)T^{2} \)
5 \( 1 + (0.216 - 0.324i)T + (-0.382 - 0.923i)T^{2} \)
11 \( 1 + (-0.923 - 0.382i)T^{2} \)
13 \( 1 + (1 + i)T + iT^{2} \)
19 \( 1 + (0.707 + 0.707i)T^{2} \)
23 \( 1 + (0.923 + 0.382i)T^{2} \)
29 \( 1 + (-1.08 + 1.63i)T + (-0.382 - 0.923i)T^{2} \)
31 \( 1 + (0.923 - 0.382i)T^{2} \)
37 \( 1 + (-0.0761 - 0.382i)T + (-0.923 + 0.382i)T^{2} \)
41 \( 1 + (0.923 + 1.38i)T + (-0.382 + 0.923i)T^{2} \)
43 \( 1 + (-0.707 + 0.707i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2} \)
59 \( 1 + (-0.707 + 0.707i)T^{2} \)
61 \( 1 + (-0.923 + 0.617i)T + (0.382 - 0.923i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (0.923 - 0.382i)T^{2} \)
73 \( 1 + (1.08 - 1.63i)T + (-0.382 - 0.923i)T^{2} \)
79 \( 1 + (-0.923 - 0.382i)T^{2} \)
83 \( 1 + (-0.707 - 0.707i)T^{2} \)
89 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
97 \( 1 + (-1.63 - 1.08i)T + (0.382 + 0.923i)T^{2} \)
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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

−8.699320653013980198974910490467, −7.924341774477703752384414606776, −7.42201550867929985063845216218, −6.76951078937064599307901175941, −5.87811139570125816300917612911, −4.79149629415506422368572906480, −3.72927509088552573284648618315, −3.05561388486647462761963313863, −2.03036142762918088964082472660, −0.805824639668741540567218698872, 1.12894925946816484229440663141, 2.09187156278542548737629572141, 3.19572815556730129968096934807, 4.68053034239972649787745198578, 4.92156409662951244539727457085, 6.14707267726628780394613229439, 6.94840511352471239980478048376, 7.39279439346930465545798021079, 8.159878161823216832288068538897, 8.873452687552963370150654233356

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