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Properties

Label 2-3332-476.387-c0-0-3
Degree $2$
Conductor $3332$
Sign $0.710 - 0.704i$
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  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (1.36 − 0.366i)5-s + 0.999i·8-s + (−0.866 − 0.5i)9-s + (1.36 + 0.366i)10-s + 2·13-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.5i)17-s + (−0.499 − 0.866i)18-s + (0.999 + 0.999i)20-s + (0.866 − 0.5i)25-s + (1.73 + i)26-s + (−1 − i)29-s + (−0.866 + 0.499i)32-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (1.36 − 0.366i)5-s + 0.999i·8-s + (−0.866 − 0.5i)9-s + (1.36 + 0.366i)10-s + 2·13-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.5i)17-s + (−0.499 − 0.866i)18-s + (0.999 + 0.999i)20-s + (0.866 − 0.5i)25-s + (1.73 + i)26-s + (−1 − i)29-s + (−0.866 + 0.499i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.710 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.710 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.710 - 0.704i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.710 - 0.704i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.628893994\)
\(L(\frac12)\) \(\approx\) \(2.628893994\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 \)
17 \( 1 + (0.866 - 0.5i)T \)
good3 \( 1 + (0.866 + 0.5i)T^{2} \)
5 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.866 - 0.5i)T^{2} \)
13 \( 1 - 2T + T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.866 - 0.5i)T^{2} \)
29 \( 1 + (1 + i)T + iT^{2} \)
31 \( 1 + (0.866 + 0.5i)T^{2} \)
37 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
41 \( 1 + (-1 + i)T - iT^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
79 \( 1 + (0.866 - 0.5i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (1 + i)T + iT^{2} \)
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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

−8.691203231426223991736343398602, −8.390995080227855423496986655323, −7.13042105437982099966483284733, −6.29297238121707521700936330877, −5.86357781332091742982085489698, −5.46136599186621778450574971078, −4.23651535896121596928196851308, −3.56252641311759928764043108461, −2.49724869189337234910806714464, −1.57587011085967825806985705991, 1.44995098844442317643185474940, 2.23577455025725957721803482243, 3.09168935815191782732982708544, 3.90134140255946095315776947519, 5.11301011010912190998462233852, 5.58710324861576035781463989279, 6.32846239736859457853336198308, 6.74459308188848597710047840130, 8.003476686237513307009532961569, 9.056078902727679139194131891467

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