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Properties

Label 2-1380-92.91-c1-0-72
Degree $2$
Conductor $1380$
Sign $-0.0890 + 0.996i$
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  + (−0.433 + 1.34i)2-s − i·3-s + (−1.62 − 1.16i)4-s − i·5-s + (1.34 + 0.433i)6-s − 3.66·7-s + (2.27 − 1.67i)8-s − 9-s + (1.34 + 0.433i)10-s + 2.17·11-s + (−1.16 + 1.62i)12-s + 6.11·13-s + (1.58 − 4.92i)14-s − 15-s + (1.27 + 3.79i)16-s + 2.05i·17-s + ⋯
L(s)  = 1  + (−0.306 + 0.951i)2-s − 0.577i·3-s + (−0.811 − 0.583i)4-s − 0.447i·5-s + (0.549 + 0.177i)6-s − 1.38·7-s + (0.804 − 0.593i)8-s − 0.333·9-s + (0.425 + 0.137i)10-s + 0.656·11-s + (−0.337 + 0.468i)12-s + 1.69·13-s + (0.424 − 1.31i)14-s − 0.258·15-s + (0.318 + 0.948i)16-s + 0.499i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0890 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0890 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.0890 + 0.996i$
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: \((2,\ 1380,\ (\ :1/2),\ -0.0890 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6907376386\)
\(L(\frac12)\) \(\approx\) \(0.6907376386\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.433 - 1.34i)T \)
3 \( 1 + iT \)
5 \( 1 + iT \)
23 \( 1 + (2.44 + 4.12i)T \)
good7 \( 1 + 3.66T + 7T^{2} \)
11 \( 1 - 2.17T + 11T^{2} \)
13 \( 1 - 6.11T + 13T^{2} \)
17 \( 1 - 2.05iT - 17T^{2} \)
19 \( 1 - 1.02T + 19T^{2} \)
29 \( 1 + 6.97T + 29T^{2} \)
31 \( 1 + 3.10iT - 31T^{2} \)
37 \( 1 + 8.95iT - 37T^{2} \)
41 \( 1 - 4.62T + 41T^{2} \)
43 \( 1 + 8.16T + 43T^{2} \)
47 \( 1 + 9.68iT - 47T^{2} \)
53 \( 1 - 5.81iT - 53T^{2} \)
59 \( 1 - 1.80iT - 59T^{2} \)
61 \( 1 + 8.11iT - 61T^{2} \)
67 \( 1 + 15.0T + 67T^{2} \)
71 \( 1 + 4.61iT - 71T^{2} \)
73 \( 1 + 4.66T + 73T^{2} \)
79 \( 1 + 12.9T + 79T^{2} \)
83 \( 1 + 10.8T + 83T^{2} \)
89 \( 1 - 4.31iT - 89T^{2} \)
97 \( 1 + 18.8iT - 97T^{2} \)
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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

−9.013047040994572800470959170641, −8.689766000883091810971027696708, −7.68058760291081553786812636600, −6.83702513093682681088213490121, −6.06215271453644198524397792318, −5.77867248974351913980452525014, −4.19951191622326624911906874356, −3.49219090721144115940343223635, −1.64524623288936010970303914408, −0.33934231833672169968590203775, 1.41446167656612567688145342584, 3.04262568553744558595364315891, 3.46681660697798622814975765136, 4.28254011930367049164346027576, 5.66238694353063777584781585273, 6.43544047281739551562750022609, 7.47305460509410405693896061667, 8.594578679227637940271158259193, 9.215421129218110234598740431783, 9.852522540082388001411192584850

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