Properties

Label 2-1380-345.344-c1-0-24
Degree $2$
Conductor $1380$
Sign $0.832 + 0.554i$
Analytic cond. $11.0193$
Root an. cond. $3.31954$
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  + (0.403 + 1.68i)3-s + (−2.20 + 0.393i)5-s − 3.85·7-s + (−2.67 + 1.36i)9-s + 3.62·11-s − 1.57i·13-s + (−1.55 − 3.54i)15-s − 5.19i·17-s − 6.52i·19-s + (−1.55 − 6.49i)21-s + (−4.03 + 2.59i)23-s + (4.69 − 1.73i)25-s + (−3.37 − 3.95i)27-s + 9.95i·29-s + 8.99·31-s + ⋯
L(s)  = 1  + (0.233 + 0.972i)3-s + (−0.984 + 0.176i)5-s − 1.45·7-s + (−0.891 + 0.453i)9-s + 1.09·11-s − 0.437i·13-s + (−0.400 − 0.916i)15-s − 1.25i·17-s − 1.49i·19-s + (−0.339 − 1.41i)21-s + (−0.841 + 0.540i)23-s + (0.938 − 0.346i)25-s + (−0.648 − 0.761i)27-s + 1.84i·29-s + 1.61·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.832 + 0.554i$
Analytic conductor: \(11.0193\)
Root analytic conductor: \(3.31954\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :1/2),\ 0.832 + 0.554i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8483022506\)
\(L(\frac12)\) \(\approx\) \(0.8483022506\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.403 - 1.68i)T \)
5 \( 1 + (2.20 - 0.393i)T \)
23 \( 1 + (4.03 - 2.59i)T \)
good7 \( 1 + 3.85T + 7T^{2} \)
11 \( 1 - 3.62T + 11T^{2} \)
13 \( 1 + 1.57iT - 13T^{2} \)
17 \( 1 + 5.19iT - 17T^{2} \)
19 \( 1 + 6.52iT - 19T^{2} \)
29 \( 1 - 9.95iT - 29T^{2} \)
31 \( 1 - 8.99T + 31T^{2} \)
37 \( 1 - 5.60T + 37T^{2} \)
41 \( 1 + 2.15iT - 41T^{2} \)
43 \( 1 + 8.84T + 43T^{2} \)
47 \( 1 - 9.35T + 47T^{2} \)
53 \( 1 - 6.55iT - 53T^{2} \)
59 \( 1 + 5.15iT - 59T^{2} \)
61 \( 1 + 10.7iT - 61T^{2} \)
67 \( 1 + 2.14T + 67T^{2} \)
71 \( 1 + 8.71iT - 71T^{2} \)
73 \( 1 + 10.6iT - 73T^{2} \)
79 \( 1 + 9.40iT - 79T^{2} \)
83 \( 1 - 5.23iT - 83T^{2} \)
89 \( 1 - 7.33T + 89T^{2} \)
97 \( 1 - 2.45T + 97T^{2} \)
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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

−9.292410285653450259659799275682, −9.089546636952474426383542302090, −7.961789534677382297676868721277, −6.97951222594482280461663637120, −6.36106159253582399098101570961, −5.08826246402263849252625607593, −4.27235744860130366327686606255, −3.30733865343944814993873250227, −2.88828232482031436370665592336, −0.40405918263531110244346759837, 1.06730575327891496927795939691, 2.49284152202761829221105644863, 3.72695518007019545129001782285, 4.10484101365058706987753788812, 6.15549887285727812266666902770, 6.20271689477240641836613619792, 7.18945184952000673172077575730, 8.164358402925254586579836429940, 8.545199066280366050590571441131, 9.644906009696546674231388405879

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