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Properties

Label 2-1140-1140.1139-c0-0-14
Degree $2$
Conductor $1140$
Sign $-0.382 + 0.923i$
Analytic cond. $0.568934$
Root an. cond. $0.754277$
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  + (−0.382 − 0.923i)2-s + (0.923 − 0.382i)3-s + (−0.707 + 0.707i)4-s − i·5-s + (−0.707 − 0.707i)6-s + (0.923 + 0.382i)8-s + (0.707 − 0.707i)9-s + (−0.923 + 0.382i)10-s + 1.41·11-s + (−0.382 + 0.923i)12-s − 0.765·13-s + (−0.382 − 0.923i)15-s − i·16-s + (−0.923 − 0.382i)18-s + i·19-s + (0.707 + 0.707i)20-s + ⋯
L(s)  = 1  + (−0.382 − 0.923i)2-s + (0.923 − 0.382i)3-s + (−0.707 + 0.707i)4-s − i·5-s + (−0.707 − 0.707i)6-s + (0.923 + 0.382i)8-s + (0.707 − 0.707i)9-s + (−0.923 + 0.382i)10-s + 1.41·11-s + (−0.382 + 0.923i)12-s − 0.765·13-s + (−0.382 − 0.923i)15-s − i·16-s + (−0.923 − 0.382i)18-s + i·19-s + (0.707 + 0.707i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1140\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.382 + 0.923i$
Analytic conductor: \(0.568934\)
Root analytic conductor: \(0.754277\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1140} (1139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1140,\ (\ :0),\ -0.382 + 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.147740618\)
\(L(\frac12)\) \(\approx\) \(1.147740618\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.382 + 0.923i)T \)
3 \( 1 + (-0.923 + 0.382i)T \)
5 \( 1 + iT \)
19 \( 1 - iT \)
good7 \( 1 + T^{2} \)
11 \( 1 - 1.41T + T^{2} \)
13 \( 1 + 0.765T + T^{2} \)
17 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + 1.84T + T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 0.765iT - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 1.41T + T^{2} \)
67 \( 1 - 1.84iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - 1.84T + T^{2} \)
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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

−9.656319771527435860285596132546, −8.950448767171326593105596664214, −8.428826513648528809360111847455, −7.61847396403176322114574632564, −6.66188777791994103085705366143, −5.19920296785766354002532137451, −4.12447827074578879891939528858, −3.50679954137876828950412931867, −2.10182215880873958527264954152, −1.26061254183124325864705538260, 1.89018879253115938504493113788, 3.24864472113092203855922848984, 4.19412725093819740160565406491, 5.14883850697741328130386332597, 6.47005646515028477531044069128, 7.00450866967546414698549262660, 7.70921197122765435987801483228, 8.707827098909734170948423446957, 9.314758383529781414095984385120, 9.973537535361212002792766982620

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