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Properties

Label 2-1140-95.18-c1-0-10
Degree $2$
Conductor $1140$
Sign $0.818 + 0.574i$
Analytic cond. $9.10294$
Root an. cond. $3.01710$
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  + (−0.707 + 0.707i)3-s + (−1.45 + 1.70i)5-s + (1.40 − 1.40i)7-s − 1.00i·9-s − 3.57·11-s + (4.12 − 4.12i)13-s + (−0.177 − 2.22i)15-s + (−3.04 + 3.04i)17-s + (0.597 − 4.31i)19-s + 1.99i·21-s + (−1.02 − 1.02i)23-s + (−0.792 − 4.93i)25-s + (0.707 + 0.707i)27-s + 6.83·29-s − 1.09i·31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (−0.648 + 0.761i)5-s + (0.532 − 0.532i)7-s − 0.333i·9-s − 1.07·11-s + (1.14 − 1.14i)13-s + (−0.0459 − 0.575i)15-s + (−0.737 + 0.737i)17-s + (0.137 − 0.990i)19-s + 0.434i·21-s + (−0.213 − 0.213i)23-s + (−0.158 − 0.987i)25-s + (0.136 + 0.136i)27-s + 1.27·29-s − 0.195i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.818 + 0.574i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.818 + 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1140\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.818 + 0.574i$
Analytic conductor: \(9.10294\)
Root analytic conductor: \(3.01710\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1140} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1140,\ (\ :1/2),\ 0.818 + 0.574i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.078665367\)
\(L(\frac12)\) \(\approx\) \(1.078665367\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (1.45 - 1.70i)T \)
19 \( 1 + (-0.597 + 4.31i)T \)
good7 \( 1 + (-1.40 + 1.40i)T - 7iT^{2} \)
11 \( 1 + 3.57T + 11T^{2} \)
13 \( 1 + (-4.12 + 4.12i)T - 13iT^{2} \)
17 \( 1 + (3.04 - 3.04i)T - 17iT^{2} \)
23 \( 1 + (1.02 + 1.02i)T + 23iT^{2} \)
29 \( 1 - 6.83T + 29T^{2} \)
31 \( 1 + 1.09iT - 31T^{2} \)
37 \( 1 + (-0.0652 - 0.0652i)T + 37iT^{2} \)
41 \( 1 + 4.54iT - 41T^{2} \)
43 \( 1 + (-0.442 - 0.442i)T + 43iT^{2} \)
47 \( 1 + (-4.93 + 4.93i)T - 47iT^{2} \)
53 \( 1 + (-4.43 + 4.43i)T - 53iT^{2} \)
59 \( 1 - 8.82T + 59T^{2} \)
61 \( 1 - 4.30T + 61T^{2} \)
67 \( 1 + (-7.18 - 7.18i)T + 67iT^{2} \)
71 \( 1 + 11.8iT - 71T^{2} \)
73 \( 1 + (1.32 + 1.32i)T + 73iT^{2} \)
79 \( 1 - 1.90T + 79T^{2} \)
83 \( 1 + (8.76 + 8.76i)T + 83iT^{2} \)
89 \( 1 - 6.98T + 89T^{2} \)
97 \( 1 + (5.68 + 5.68i)T + 97iT^{2} \)
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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

−10.18942383050917145569762132485, −8.675847914630755109457016158417, −8.136765271751951351262204883826, −7.26053362081120868157705004239, −6.39091471331100363243334523624, −5.43278076269055944107830813294, −4.45739180644429213391380810309, −3.60387337278202276017165834365, −2.54660682762141368321580006027, −0.58484334370908014429349816365, 1.15704764335117058330456785959, 2.38715664480929631454990566025, 3.90336539642990450918153458988, 4.81548173466727493549129145531, 5.55140992169106184198089524323, 6.53396945607101166513068260217, 7.51602866824800123566587909281, 8.348955058866137588760666848823, 8.765518293581894080819596603867, 9.871965961464503235091721591652

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