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Properties

Label 2-3360-40.29-c1-0-24
Degree $2$
Conductor $3360$
Sign $0.660 - 0.751i$
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  − 3-s + (−0.725 + 2.11i)5-s − i·7-s + 9-s − 1.98i·11-s + 4.46·13-s + (0.725 − 2.11i)15-s + 6.83i·17-s − 6.40i·19-s + i·21-s + 8.27i·23-s + (−3.94 − 3.06i)25-s − 27-s − 2.08i·29-s − 1.28·31-s + ⋯
L(s)  = 1  − 0.577·3-s + (−0.324 + 0.945i)5-s − 0.377i·7-s + 0.333·9-s − 0.598i·11-s + 1.23·13-s + (0.187 − 0.546i)15-s + 1.65i·17-s − 1.46i·19-s + 0.218i·21-s + 1.72i·23-s + (−0.789 − 0.613i)25-s − 0.192·27-s − 0.386i·29-s − 0.230·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.660 - 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.660 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3360\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.660 - 0.751i$
Analytic conductor: \(26.8297\)
Root analytic conductor: \(5.17974\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3360} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3360,\ (\ :1/2),\ 0.660 - 0.751i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.386759051\)
\(L(\frac12)\) \(\approx\) \(1.386759051\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 + (0.725 - 2.11i)T \)
7 \( 1 + iT \)
good11 \( 1 + 1.98iT - 11T^{2} \)
13 \( 1 - 4.46T + 13T^{2} \)
17 \( 1 - 6.83iT - 17T^{2} \)
19 \( 1 + 6.40iT - 19T^{2} \)
23 \( 1 - 8.27iT - 23T^{2} \)
29 \( 1 + 2.08iT - 29T^{2} \)
31 \( 1 + 1.28T + 31T^{2} \)
37 \( 1 - 10.5T + 37T^{2} \)
41 \( 1 + 5.84T + 41T^{2} \)
43 \( 1 - 0.807T + 43T^{2} \)
47 \( 1 + 9.01iT - 47T^{2} \)
53 \( 1 - 4.80T + 53T^{2} \)
59 \( 1 + 1.35iT - 59T^{2} \)
61 \( 1 + 11.1iT - 61T^{2} \)
67 \( 1 + 6.01T + 67T^{2} \)
71 \( 1 - 7.23T + 71T^{2} \)
73 \( 1 - 0.829iT - 73T^{2} \)
79 \( 1 - 11.0T + 79T^{2} \)
83 \( 1 + 2.69T + 83T^{2} \)
89 \( 1 - 2.66T + 89T^{2} \)
97 \( 1 - 0.641iT - 97T^{2} \)
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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

−8.605214801266393757079306959016, −7.901194617032717043771226974875, −7.17226274856492328131126783911, −6.31292806385036234605615072898, −6.00718390740896685152274359227, −4.93672856689053791408721983567, −3.73714113145313496770202833939, −3.53782115918657160294191861906, −2.12605336007258219204188341996, −0.876509388838645952953539352493, 0.63516919718752185027520886585, 1.64594014974138246647305280416, 2.91386880740911566620422730395, 4.12646778847606997720234088556, 4.62629709545478642618376627859, 5.52235360799772662628378553724, 6.11372219625181054122542079749, 7.01056486559846173093961435407, 7.86792994346925268993164622227, 8.504725657647815923100889998321

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