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Properties

Label 2-1380-345.344-c1-0-8
Degree $2$
Conductor $1380$
Sign $-0.552 - 0.833i$
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  + (−1.71 − 0.253i)3-s + (0.923 + 2.03i)5-s + 2.63·7-s + (2.87 + 0.870i)9-s − 4.99·11-s + 4.22i·13-s + (−1.06 − 3.72i)15-s + 4.76i·17-s − 1.53i·19-s + (−4.52 − 0.670i)21-s + (4.57 − 1.44i)23-s + (−3.29 + 3.76i)25-s + (−4.69 − 2.22i)27-s − 0.927i·29-s + 1.66·31-s + ⋯
L(s)  = 1  + (−0.989 − 0.146i)3-s + (0.412 + 0.910i)5-s + 0.997·7-s + (0.956 + 0.290i)9-s − 1.50·11-s + 1.17i·13-s + (−0.274 − 0.961i)15-s + 1.15i·17-s − 0.352i·19-s + (−0.986 − 0.146i)21-s + (0.953 − 0.301i)23-s + (−0.658 + 0.752i)25-s + (−0.904 − 0.427i)27-s − 0.172i·29-s + 0.299·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.552 - 0.833i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.552 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.552 - 0.833i$
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: \((2,\ 1380,\ (\ :1/2),\ -0.552 - 0.833i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9519696277\)
\(L(\frac12)\) \(\approx\) \(0.9519696277\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.71 + 0.253i)T \)
5 \( 1 + (-0.923 - 2.03i)T \)
23 \( 1 + (-4.57 + 1.44i)T \)
good7 \( 1 - 2.63T + 7T^{2} \)
11 \( 1 + 4.99T + 11T^{2} \)
13 \( 1 - 4.22iT - 13T^{2} \)
17 \( 1 - 4.76iT - 17T^{2} \)
19 \( 1 + 1.53iT - 19T^{2} \)
29 \( 1 + 0.927iT - 29T^{2} \)
31 \( 1 - 1.66T + 31T^{2} \)
37 \( 1 + 7.06T + 37T^{2} \)
41 \( 1 + 11.2iT - 41T^{2} \)
43 \( 1 - 0.983T + 43T^{2} \)
47 \( 1 + 8.19T + 47T^{2} \)
53 \( 1 - 10.8iT - 53T^{2} \)
59 \( 1 - 4.23iT - 59T^{2} \)
61 \( 1 - 12.0iT - 61T^{2} \)
67 \( 1 - 5.05T + 67T^{2} \)
71 \( 1 - 10.6iT - 71T^{2} \)
73 \( 1 + 11.4iT - 73T^{2} \)
79 \( 1 + 0.925iT - 79T^{2} \)
83 \( 1 - 12.6iT - 83T^{2} \)
89 \( 1 + 2.70T + 89T^{2} \)
97 \( 1 + 4.80T + 97T^{2} \)
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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

−10.29264996516866290614277475973, −9.092799610794817918086039383846, −8.073726630754311194557072868809, −7.25163608368232242922755595258, −6.62782387465073050541503993666, −5.65462909573179969895009619805, −5.01845250828843615211915652391, −4.04314322768486490102960871734, −2.52569190326664617795255937064, −1.58786032570079227583028809180, 0.44307928719845982026295121391, 1.66357844126490645977036777587, 3.10910342881512889312741259167, 4.75601878217130051496148942165, 5.08439805522438429537358429362, 5.58626932833586495539589516664, 6.79658361042107653225910836176, 7.88422676802941662533995018301, 8.264523832036349131659800863477, 9.547377219082500095396026469463

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