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Properties

Label 2-3360-840.293-c0-0-6
Degree $2$
Conductor $3360$
Sign $0.850 + 0.525i$
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  + (0.923 − 0.382i)3-s + (0.382 − 0.923i)5-s + (−0.707 + 0.707i)7-s + (0.707 − 0.707i)9-s + (1.30 + 1.30i)13-s − i·15-s + 0.765i·19-s + (−0.382 + 0.923i)21-s + (1 − i)23-s + (−0.707 − 0.707i)25-s + (0.382 − 0.923i)27-s + (0.382 + 0.923i)35-s + (1.70 + 0.707i)39-s + (−0.382 − 0.923i)45-s − 1.00i·49-s + ⋯
L(s)  = 1  + (0.923 − 0.382i)3-s + (0.382 − 0.923i)5-s + (−0.707 + 0.707i)7-s + (0.707 − 0.707i)9-s + (1.30 + 1.30i)13-s − i·15-s + 0.765i·19-s + (−0.382 + 0.923i)21-s + (1 − i)23-s + (−0.707 − 0.707i)25-s + (0.382 − 0.923i)27-s + (0.382 + 0.923i)35-s + (1.70 + 0.707i)39-s + (−0.382 − 0.923i)45-s − 1.00i·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3360\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.850 + 0.525i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3360} (1553, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3360,\ (\ :0),\ 0.850 + 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.890282692\)
\(L(\frac12)\) \(\approx\) \(1.890282692\)
\(L(1)\) 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 \( 1 + (-0.923 + 0.382i)T \)
5 \( 1 + (-0.382 + 0.923i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
good11 \( 1 + T^{2} \)
13 \( 1 + (-1.30 - 1.30i)T + iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 - 0.765iT - T^{2} \)
23 \( 1 + (-1 + i)T - iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + 0.765T + T^{2} \)
61 \( 1 + 1.84T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - 1.41iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + 1.41iT - T^{2} \)
83 \( 1 + (-0.541 + 0.541i)T - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + iT^{2} \)
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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

−8.779026530438671348160263882324, −8.316178089836666851715382786958, −7.27387283630771954201121122538, −6.35169492290710453184226359616, −6.01703792033145970183654980520, −4.78485461870694382978351848608, −3.99127172650179402917495897034, −3.11929513694387271937861152962, −2.09214613025908790728227739295, −1.27639107067994934020706986345, 1.34653644884919302890863588244, 2.74880359890841456435023813024, 3.26654607571205405692356201065, 3.85146029536737702834516317810, 5.01750723338593252032394285788, 5.96598975432454047995779389500, 6.72537048746637882453139101025, 7.46663453508364212080486827178, 8.017308285759532107682197420120, 9.058323323502876587414354737213

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