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Properties

Label 2-3360-12.11-c1-0-30
Degree $2$
Conductor $3360$
Sign $-0.357 - 0.933i$
Analytic cond. $26.8297$
Root an. cond. $5.17974$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.705 + 1.58i)3-s + i·5-s − i·7-s + (−2.00 − 2.23i)9-s + 0.590·11-s + 0.402·13-s + (−1.58 − 0.705i)15-s − 3.70i·17-s + 4.77i·19-s + (1.58 + 0.705i)21-s + 3.55·23-s − 25-s + (4.94 − 1.59i)27-s + 0.729i·29-s − 0.453i·31-s + ⋯
L(s)  = 1  + (−0.407 + 0.913i)3-s + 0.447i·5-s − 0.377i·7-s + (−0.668 − 0.743i)9-s + 0.178·11-s + 0.111·13-s + (−0.408 − 0.182i)15-s − 0.897i·17-s + 1.09i·19-s + (0.345 + 0.153i)21-s + 0.741·23-s − 0.200·25-s + (0.951 − 0.307i)27-s + 0.135i·29-s − 0.0814i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.357 - 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.357 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3360\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.357 - 0.933i$
Analytic conductor: \(26.8297\)
Root analytic conductor: \(5.17974\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3360} (2591, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3360,\ (\ :1/2),\ -0.357 - 0.933i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.306228302\)
\(L(\frac12)\) \(\approx\) \(1.306228302\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.705 - 1.58i)T \)
5 \( 1 - iT \)
7 \( 1 + iT \)
good11 \( 1 - 0.590T + 11T^{2} \)
13 \( 1 - 0.402T + 13T^{2} \)
17 \( 1 + 3.70iT - 17T^{2} \)
19 \( 1 - 4.77iT - 19T^{2} \)
23 \( 1 - 3.55T + 23T^{2} \)
29 \( 1 - 0.729iT - 29T^{2} \)
31 \( 1 + 0.453iT - 31T^{2} \)
37 \( 1 + 0.804T + 37T^{2} \)
41 \( 1 - 0.972iT - 41T^{2} \)
43 \( 1 - 3.35iT - 43T^{2} \)
47 \( 1 - 1.11T + 47T^{2} \)
53 \( 1 - 8.65iT - 53T^{2} \)
59 \( 1 - 4.77T + 59T^{2} \)
61 \( 1 - 2.59T + 61T^{2} \)
67 \( 1 - 10.2iT - 67T^{2} \)
71 \( 1 - 6.78T + 71T^{2} \)
73 \( 1 - 2.81T + 73T^{2} \)
79 \( 1 - 17.3iT - 79T^{2} \)
83 \( 1 - 5.10T + 83T^{2} \)
89 \( 1 + 10.1iT - 89T^{2} \)
97 \( 1 + 7.16T + 97T^{2} \)
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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

−8.995874842739374879552172999987, −8.163091103655217228866439627901, −7.26413773823610237413978959504, −6.56290681664808239733423936225, −5.75494011921261320631373115733, −5.03475788882525996399209921185, −4.16070613355202834678284178332, −3.47561330083119826761707578129, −2.58905426529818339825254904531, −1.02835448708551532017879317286, 0.50058453856951638623245762487, 1.62611812846909474693198511845, 2.50738319355934812796432748707, 3.61775596095778400032400023136, 4.79411727547138713700275747644, 5.37372872640855298280159044047, 6.23949820790065906712482551213, 6.82034738765545041240877874092, 7.61717648797726510386539941698, 8.430975367405469621873838518274

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