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Properties

Label 2-3332-68.67-c0-0-10
Degree $2$
Conductor $3332$
Sign $-0.951 + 0.309i$
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  + (−0.809 − 0.587i)2-s − 1.17·3-s + (0.309 + 0.951i)4-s − 0.618i·5-s + (0.951 + 0.690i)6-s + (0.309 − 0.951i)8-s + 0.381·9-s + (−0.363 + 0.500i)10-s + (−0.363 − 1.11i)12-s + 0.726i·15-s + (−0.809 + 0.587i)16-s − i·17-s + (−0.309 − 0.224i)18-s + (0.587 − 0.190i)20-s + (−0.363 + 1.11i)24-s + 0.618·25-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s − 1.17·3-s + (0.309 + 0.951i)4-s − 0.618i·5-s + (0.951 + 0.690i)6-s + (0.309 − 0.951i)8-s + 0.381·9-s + (−0.363 + 0.500i)10-s + (−0.363 − 1.11i)12-s + 0.726i·15-s + (−0.809 + 0.587i)16-s − i·17-s + (−0.309 − 0.224i)18-s + (0.587 − 0.190i)20-s + (−0.363 + 1.11i)24-s + 0.618·25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 + 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 + 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-0.951 + 0.309i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ -0.951 + 0.309i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3077708090\)
\(L(\frac12)\) \(\approx\) \(0.3077708090\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 + 0.587i)T \)
7 \( 1 \)
17 \( 1 + iT \)
good3 \( 1 + 1.17T + T^{2} \)
5 \( 1 + 0.618iT - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + 1.90T + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + 1.61iT - T^{2} \)
43 \( 1 - 1.17iT - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 0.618T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 1.61iT - T^{2} \)
67 \( 1 + 1.90iT - T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + 1.61iT - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + 0.618iT - T^{2} \)
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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

−8.814300042171062404909868868649, −7.69434771544947194306194902520, −7.11888323588506638636820193024, −6.27304439880292617869404340088, −5.36664326617980161006425034010, −4.73141434419703789289368880232, −3.71170158658171454974450518195, −2.63665263051126010664007305395, −1.42701142835708391599590223461, −0.31735805659882902828967360592, 1.25850729758525555311915982887, 2.46470683101953460461444464079, 3.77635662762562764657124836406, 4.96990524842218432929901397662, 5.57888662809335380580216913149, 6.27306220030243597255375765618, 6.82093458058479912367757005177, 7.49976586337010812468821917773, 8.402679133627730379142961217648, 9.038870615576231985817503284027

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