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Properties

Label 2-1380-92.91-c1-0-67
Degree $2$
Conductor $1380$
Sign $-0.277 + 0.960i$
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  + (−1.25 − 0.660i)2-s − i·3-s + (1.12 + 1.65i)4-s + i·5-s + (−0.660 + 1.25i)6-s + 3.91·7-s + (−0.319 − 2.81i)8-s − 9-s + (0.660 − 1.25i)10-s + 3.56·11-s + (1.65 − 1.12i)12-s − 6.21·13-s + (−4.89 − 2.58i)14-s + 15-s + (−1.45 + 3.72i)16-s − 6.87i·17-s + ⋯
L(s)  = 1  + (−0.884 − 0.466i)2-s − 0.577i·3-s + (0.563 + 0.825i)4-s + 0.447i·5-s + (−0.269 + 0.510i)6-s + 1.47·7-s + (−0.113 − 0.993i)8-s − 0.333·9-s + (0.208 − 0.395i)10-s + 1.07·11-s + (0.476 − 0.325i)12-s − 1.72·13-s + (−1.30 − 0.690i)14-s + 0.258·15-s + (−0.363 + 0.931i)16-s − 1.66i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.277 + 0.960i$
Analytic conductor: \(11.0193\)
Root analytic conductor: \(3.31954\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :1/2),\ -0.277 + 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.094026299\)
\(L(\frac12)\) \(\approx\) \(1.094026299\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.25 + 0.660i)T \)
3 \( 1 + iT \)
5 \( 1 - iT \)
23 \( 1 + (3.05 + 3.69i)T \)
good7 \( 1 - 3.91T + 7T^{2} \)
11 \( 1 - 3.56T + 11T^{2} \)
13 \( 1 + 6.21T + 13T^{2} \)
17 \( 1 + 6.87iT - 17T^{2} \)
19 \( 1 + 5.29T + 19T^{2} \)
29 \( 1 - 7.14T + 29T^{2} \)
31 \( 1 + 5.49iT - 31T^{2} \)
37 \( 1 - 1.30iT - 37T^{2} \)
41 \( 1 - 10.4T + 41T^{2} \)
43 \( 1 + 3.43T + 43T^{2} \)
47 \( 1 + 2.84iT - 47T^{2} \)
53 \( 1 + 11.0iT - 53T^{2} \)
59 \( 1 + 5.12iT - 59T^{2} \)
61 \( 1 + 5.14iT - 61T^{2} \)
67 \( 1 - 14.0T + 67T^{2} \)
71 \( 1 - 11.6iT - 71T^{2} \)
73 \( 1 - 11.4T + 73T^{2} \)
79 \( 1 - 0.545T + 79T^{2} \)
83 \( 1 + 3.08T + 83T^{2} \)
89 \( 1 + 7.41iT - 89T^{2} \)
97 \( 1 - 7.95iT - 97T^{2} \)
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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

−9.421243951259631410370721077281, −8.388359948607076023267447275303, −7.896868481712701094638344762416, −7.03452233212380771636407924458, −6.52136144639899915381146195772, −5.00642939489446601405948851898, −4.14634425584382418465589246692, −2.56085947167197228272786514248, −2.07832508480241162109909306703, −0.62671603463529920842932774874, 1.34993891705800116491316657816, 2.28802339308224979598925580296, 4.17037933689412582803209892304, 4.78220100812773527318108398080, 5.71158890427382603862696424379, 6.61344770053250620555402430946, 7.69975353891742101895237789782, 8.274666830033320574465358652243, 8.922150773371696450315815614386, 9.691238003001563690967168097957

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