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Properties

Label 2-3332-68.47-c0-0-2
Degree $2$
Conductor $3332$
Sign $-0.615 + 0.788i$
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 + 2·13-s + 16-s − i·17-s + 18-s + (1 + i)20-s + i·25-s − 2i·26-s + (−1 − i)29-s − i·32-s + ⋯
L(s)  = 1  − i·2-s − 4-s + (−1 − i)5-s + i·8-s + i·9-s + (−1 + i)10-s + 2·13-s + 16-s − i·17-s + 18-s + (1 + i)20-s + i·25-s − 2i·26-s + (−1 − i)29-s − i·32-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-0.615 + 0.788i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (3039, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ -0.615 + 0.788i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9410712447\)
\(L(\frac12)\) \(\approx\) \(0.9410712447\)
\(L(1)\) 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 \( 1 + iT \)
good3 \( 1 - iT^{2} \)
5 \( 1 + (1 + i)T + iT^{2} \)
11 \( 1 + iT^{2} \)
13 \( 1 - 2T + T^{2} \)
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 \( 1 + 2T + T^{2} \)
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

−8.548549548274563564144906219624, −8.098908181856115567616957731689, −7.41986238014984211437602090300, −6.00542313895195228226626409227, −5.26087062239580847145273039016, −4.40871171970737961750501582965, −3.94905446137667590140928639264, −2.96896215101482196034074936860, −1.77697439988707306132956663952, −0.71656054501944727655211663883, 1.15790871148051782974946018628, 3.11611928451943545657526856909, 3.85246361645720185831238690570, 4.13088693244731173451210324301, 5.69325487277978777709354685503, 6.15038956349376653971524218849, 6.84314331405163844154461594750, 7.48358014587181380502623646717, 8.274342717562745769936631149975, 8.790390094214946818624091041978

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