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Properties

Label 2-1380-1380.1379-c0-0-1
Degree $2$
Conductor $1380$
Sign $-i$
Analytic cond. $0.688709$
Root an. cond. $0.829885$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s + i·3-s − 4-s + i·5-s + 6-s + i·8-s − 9-s + 10-s − i·12-s − 15-s + 16-s + 2i·17-s + i·18-s − 2·19-s − i·20-s + ⋯
L(s)  = 1  − i·2-s + i·3-s − 4-s + i·5-s + 6-s + i·8-s − 9-s + 10-s − i·12-s − 15-s + 16-s + 2i·17-s + i·18-s − 2·19-s − i·20-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-i$
Analytic conductor: \(0.688709\)
Root analytic conductor: \(0.829885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (1379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :0),\ -i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7001407118\)
\(L(\frac12)\) \(\approx\) \(0.7001407118\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 - iT \)
5 \( 1 - iT \)
23 \( 1 + iT \)
good7 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - 2iT - T^{2} \)
19 \( 1 + 2T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - 2iT - T^{2} \)
53 \( 1 - 2iT - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - 2T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + T^{2} \)
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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

−10.36813686835224664494746684207, −9.369533638429744423863950124222, −8.577435238272683510058874830952, −7.958918134294068022676806683734, −6.38186312534758947347552664086, −5.83538437261250547814978876790, −4.38790267255842035621819589650, −4.03380675684947779206408250966, −2.95940106074318793511277975517, −2.07482037714421671714319013546, 0.57044828164243985179954837895, 2.11518142313447569521833670426, 3.69514779160506568788115983853, 4.88384398740486453366720759633, 5.44797679924744034403070465620, 6.43852295084310622025288788870, 7.13077934985328505126472330974, 7.899122979177183975869243675090, 8.644166479630127959406235237554, 9.141000956428058197298251015093

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