Properties

Label 2-3360-105.44-c0-0-6
Degree $2$
Conductor $3360$
Sign $0.553 + 0.832i$
Analytic cond. $1.67685$
Root an. cond. $1.29493$
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  + (0.707 − 0.707i)3-s + (0.866 + 0.5i)5-s + (0.258 − 0.965i)7-s − 1.00i·9-s + (0.965 − 0.258i)15-s + (−0.500 − 0.866i)21-s + (0.258 − 0.448i)23-s + (0.499 + 0.866i)25-s + (−0.707 − 0.707i)27-s + 1.73i·29-s + (0.707 − 0.707i)35-s i·41-s − 1.93i·43-s + (0.500 − 0.866i)45-s + (−0.707 + 1.22i)47-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)3-s + (0.866 + 0.5i)5-s + (0.258 − 0.965i)7-s − 1.00i·9-s + (0.965 − 0.258i)15-s + (−0.500 − 0.866i)21-s + (0.258 − 0.448i)23-s + (0.499 + 0.866i)25-s + (−0.707 − 0.707i)27-s + 1.73i·29-s + (0.707 − 0.707i)35-s i·41-s − 1.93i·43-s + (0.500 − 0.866i)45-s + (−0.707 + 1.22i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3360\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.553 + 0.832i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3360} (1409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3360,\ (\ :0),\ 0.553 + 0.832i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.936969251\)
\(L(\frac12)\) \(\approx\) \(1.936969251\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (-0.258 + 0.965i)T \)
good11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.258 + 0.448i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - 1.73iT - T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + iT - T^{2} \)
43 \( 1 + 1.93iT - T^{2} \)
47 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.448 - 0.258i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - 1.93T + T^{2} \)
89 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.818052349504061120590683876352, −7.77802920864979150783865623364, −7.15212713073435000062166946640, −6.69867987506764143152044779472, −5.83186993850269227713097328122, −4.87514563473481600312627489616, −3.77116944340870452643935491305, −3.03193268549033210467863904678, −2.05588211203471848073006824844, −1.18001691211894767998804880168, 1.64990733825982099400825637658, 2.44449803557646173338612020518, 3.26094377914878128619017677329, 4.46432520084043131354822745646, 5.00224437755434843561766955245, 5.80173019865100834177135592628, 6.47320596543721762544148666502, 7.82209657454645100709361537179, 8.250482650699014809978511718117, 9.019251853475402404479289638055

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