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Properties

Label 2-1320-165.164-c1-0-54
Degree $2$
Conductor $1320$
Sign $0.992 - 0.126i$
Analytic cond. $10.5402$
Root an. cond. $3.24657$
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  + (1.72 − 0.203i)3-s + (0.910 + 2.04i)5-s + 4.06·7-s + (2.91 − 0.698i)9-s + (−3.04 − 1.32i)11-s + 0.487·13-s + (1.98 + 3.32i)15-s − 6.71i·17-s − 1.77i·19-s + (6.99 − 0.826i)21-s + 4.87·23-s + (−3.34 + 3.71i)25-s + (4.87 − 1.79i)27-s − 1.32·29-s + 6.21·31-s + ⋯
L(s)  = 1  + (0.993 − 0.117i)3-s + (0.407 + 0.913i)5-s + 1.53·7-s + (0.972 − 0.232i)9-s + (−0.916 − 0.399i)11-s + 0.135·13-s + (0.511 + 0.859i)15-s − 1.62i·17-s − 0.407i·19-s + (1.52 − 0.180i)21-s + 1.01·23-s + (−0.668 + 0.743i)25-s + (0.938 − 0.345i)27-s − 0.246·29-s + 1.11·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1320\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 11\)
Sign: $0.992 - 0.126i$
Analytic conductor: \(10.5402\)
Root analytic conductor: \(3.24657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1320} (329, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1320,\ (\ :1/2),\ 0.992 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.977883957\)
\(L(\frac12)\) \(\approx\) \(2.977883957\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 + 0.203i)T \)
5 \( 1 + (-0.910 - 2.04i)T \)
11 \( 1 + (3.04 + 1.32i)T \)
good7 \( 1 - 4.06T + 7T^{2} \)
13 \( 1 - 0.487T + 13T^{2} \)
17 \( 1 + 6.71iT - 17T^{2} \)
19 \( 1 + 1.77iT - 19T^{2} \)
23 \( 1 - 4.87T + 23T^{2} \)
29 \( 1 + 1.32T + 29T^{2} \)
31 \( 1 - 6.21T + 31T^{2} \)
37 \( 1 - 7.96iT - 37T^{2} \)
41 \( 1 + 5.07T + 41T^{2} \)
43 \( 1 + 4.68T + 43T^{2} \)
47 \( 1 + 10.7T + 47T^{2} \)
53 \( 1 + 9.51T + 53T^{2} \)
59 \( 1 - 5.13iT - 59T^{2} \)
61 \( 1 - 2.61iT - 61T^{2} \)
67 \( 1 - 8.52iT - 67T^{2} \)
71 \( 1 - 6.00iT - 71T^{2} \)
73 \( 1 + 0.708T + 73T^{2} \)
79 \( 1 - 7.16iT - 79T^{2} \)
83 \( 1 - 13.8iT - 83T^{2} \)
89 \( 1 + 15.9iT - 89T^{2} \)
97 \( 1 - 0.844iT - 97T^{2} \)
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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

−9.694035526353336684749281536686, −8.663701727627427932844919699693, −8.066325545057100459082342414476, −7.32369485951270962130163637021, −6.63649224408469809544155196354, −5.22965279647639276982543963479, −4.65027870526539512461500637968, −3.13165882485486115637200366982, −2.60895205599586596859544362715, −1.41328613146511827111410756954, 1.49692026374469220654971273496, 2.07953053340908889368031515440, 3.53986908222113814314069795054, 4.70054710770112348179683848917, 5.02221376412438296555893358351, 6.26905555970846494764040083940, 7.62708225477786462378667109906, 8.153077546270276574401168206424, 8.583331380988151939630607481069, 9.498907304433052588659740169722

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