Properties

Label 2-3332-68.55-c0-0-1
Degree $2$
Conductor $3332$
Sign $-0.788 + 0.615i$
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  i·2-s − 4-s + (1 − i)5-s + i·8-s i·9-s + (−1 − i)10-s + 16-s + 17-s − 18-s + (−1 + i)20-s i·25-s + (1 − i)29-s i·32-s i·34-s + i·36-s + (−1 + i)37-s + ⋯
L(s)  = 1  i·2-s − 4-s + (1 − i)5-s + i·8-s i·9-s + (−1 − i)10-s + 16-s + 17-s − 18-s + (−1 + i)20-s i·25-s + (1 − i)29-s i·32-s i·34-s + i·36-s + (−1 + i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-0.788 + 0.615i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (2843, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ -0.788 + 0.615i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.358232211\)
\(L(\frac12)\) \(\approx\) \(1.358232211\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−8.697110020316634563007092859195, −8.218461130640259915870701265046, −6.97185243947968735825637832093, −5.94295530227994839951083086231, −5.40497688836295314121031880538, −4.60127189344037187120258141629, −3.71589776636210439291140395299, −2.82000327023422200456686663783, −1.70607229695763933127210759749, −0.894378655439629435642484357070, 1.58963277607082267870183528224, 2.76639974321406096970467182040, 3.63283968892761488034441673367, 4.90901771012926610417118480275, 5.37516842637644558425263657366, 6.21055863605209663476274700992, 6.81343652147486035026172025044, 7.50575887639937940949158821573, 8.202334424155764257956765858584, 8.988317510774996668541736067437

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