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Properties

Label 2-3360-40.29-c1-0-51
Degree $2$
Conductor $3360$
Sign $0.948 + 0.316i$
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 + (2 − i)5-s − i·7-s + 9-s + 6·13-s + (2 − i)15-s − 2i·17-s + 4i·19-s − i·21-s + 4i·23-s + (3 − 4i)25-s + 27-s + 6i·29-s + 8·31-s + (−1 − 2i)35-s + ⋯
L(s)  = 1  + 0.577·3-s + (0.894 − 0.447i)5-s − 0.377i·7-s + 0.333·9-s + 1.66·13-s + (0.516 − 0.258i)15-s − 0.485i·17-s + 0.917i·19-s − 0.218i·21-s + 0.834i·23-s + (0.600 − 0.800i)25-s + 0.192·27-s + 1.11i·29-s + 1.43·31-s + (−0.169 − 0.338i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3360\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.948 + 0.316i$
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: \((2,\ 3360,\ (\ :1/2),\ 0.948 + 0.316i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.158343537\)
\(L(\frac12)\) \(\approx\) \(3.158343537\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + i)T \)
7 \( 1 + iT \)
good11 \( 1 - 11T^{2} \)
13 \( 1 - 6T + 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 - 6iT - 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 - 2iT - 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 6iT - 59T^{2} \)
61 \( 1 + 10iT - 61T^{2} \)
67 \( 1 + 2T + 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 2iT - 97T^{2} \)
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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

−8.441411168293010299692553632695, −8.182357158335574345907535022590, −7.00916532230518471348308326378, −6.35979175866179464745774511205, −5.57579470116411348658055023530, −4.76306966282468476764715853463, −3.75845353347607572359956385098, −3.07809411424618637920148144050, −1.79951009396895517506081376126, −1.12377073257572614649069469309, 1.14962749118117861696458884888, 2.22080543800085739800945322341, 2.93719853611986329628123122159, 3.86079552578928982012256439025, 4.80713933950394354196722973975, 5.85410286136060566975694943057, 6.37530601438608607827372528320, 7.01742851937069154793296378618, 8.222697929943080246071652705327, 8.581560827924849228781922146356

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