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Properties

Label 2-1380-345.344-c1-0-31
Degree $2$
Conductor $1380$
Sign $0.490 + 0.871i$
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 + (−0.225 + 3.86i)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 + (−0.0583 + 0.998i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.490 + 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.490 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.490 + 0.871i$
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: \((2,\ 1380,\ (\ :1/2),\ 0.490 + 0.871i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9022460843\)
\(L(\frac12)\) \(\approx\) \(0.9022460843\)
\(L(\frac{3}{2})\) 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

−9.812928603737421413988730717077, −8.831314385656855898429085231698, −8.080044696111809391574765370765, −6.49191744305175655595427243989, −6.25158498845476949151590026522, −5.22135891532007138319993395121, −4.48445111595390202777532023552, −3.14284860252879499412609670554, −2.57536217296485331966678535666, −0.38383607437996795146157922899, 1.30255279789044633003555288175, 2.65029495686367506193669514019, 3.17469504982640733342531541000, 5.03396246609809194616250191180, 5.69777037018310808097027924380, 6.57655491617995294928806327121, 6.99611145558506734308556598243, 7.994740635478034163716546062225, 8.991837847604790689684914059740, 9.786653156349558224928565453961

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