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Properties

Label 2-1380-345.344-c1-0-21
Degree $2$
Conductor $1380$
Sign $0.588 - 0.808i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.588 - 0.808i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.588 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.588 - 0.808i$
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: \((2,\ 1380,\ (\ :1/2),\ 0.588 - 0.808i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.593064878\)
\(L(\frac12)\) \(\approx\) \(1.593064878\)
\(L(\frac{3}{2})\) 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

−9.761557941956066885494911738684, −8.852178473090264614970326187961, −8.117423835196654049246865086268, −7.50354772267523330368423499161, −6.31878229822145064727554108698, −5.30769184914968998176023489520, −4.49106529802904367365034468927, −3.92478030500871495303881704838, −2.81332132021290862202308321449, −1.03575290616788171842492109639, 0.949855002449548279553574035639, 1.87394914801017900207576905161, 3.32251386138248775268607999707, 4.56975648167490546888486441701, 5.06044603848174726568038832890, 6.50216807644365419990187656943, 6.97012616191040506972849205718, 7.88168575655219085008943752316, 8.588356840859990978765775399986, 8.962710491799599050530739510554

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