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Properties

Label 2-1155-77.76-c1-0-30
Degree $2$
Conductor $1155$
Sign $-0.920 - 0.390i$
Analytic cond. $9.22272$
Root an. cond. $3.03689$
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  + 2.60i·2-s + i·3-s − 4.77·4-s − i·5-s − 2.60·6-s + (2.60 + 0.474i)7-s − 7.22i·8-s − 9-s + 2.60·10-s + (3.23 + 0.726i)11-s − 4.77i·12-s + 6.07·13-s + (−1.23 + 6.77i)14-s + 15-s + 9.24·16-s + 4.27·17-s + ⋯
L(s)  = 1  + 1.84i·2-s + 0.577i·3-s − 2.38·4-s − 0.447i·5-s − 1.06·6-s + (0.983 + 0.179i)7-s − 2.55i·8-s − 0.333·9-s + 0.823·10-s + (0.975 + 0.219i)11-s − 1.37i·12-s + 1.68·13-s + (−0.330 + 1.81i)14-s + 0.258·15-s + 2.31·16-s + 1.03·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 - 0.390i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.920 - 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.920 - 0.390i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ -0.920 - 0.390i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.790543720\)
\(L(\frac12)\) \(\approx\) \(1.790543720\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
5 \( 1 + iT \)
7 \( 1 + (-2.60 - 0.474i)T \)
11 \( 1 + (-3.23 - 0.726i)T \)
good2 \( 1 - 2.60iT - 2T^{2} \)
13 \( 1 - 6.07T + 13T^{2} \)
17 \( 1 - 4.27T + 17T^{2} \)
19 \( 1 + 2.82T + 19T^{2} \)
23 \( 1 - 5.77T + 23T^{2} \)
29 \( 1 + 6.49iT - 29T^{2} \)
31 \( 1 - 2.66iT - 31T^{2} \)
37 \( 1 - 2.66T + 37T^{2} \)
41 \( 1 + 8.37T + 41T^{2} \)
43 \( 1 + 11.9iT - 43T^{2} \)
47 \( 1 - 12.6iT - 47T^{2} \)
53 \( 1 + 13.1T + 53T^{2} \)
59 \( 1 + 4.89iT - 59T^{2} \)
61 \( 1 - 7.86T + 61T^{2} \)
67 \( 1 - 2.92T + 67T^{2} \)
71 \( 1 + 11.1T + 71T^{2} \)
73 \( 1 - 4.53T + 73T^{2} \)
79 \( 1 - 7.42iT - 79T^{2} \)
83 \( 1 + 9.31T + 83T^{2} \)
89 \( 1 + 0.418iT - 89T^{2} \)
97 \( 1 + 2.43iT - 97T^{2} \)
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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

−9.693016923084191302546133306929, −8.929168323656434275675118625285, −8.455131552215748051769057576619, −7.81008864648184783865806804150, −6.70659454299442195682026103354, −5.96176395478311910955976560782, −5.20908262729124926643187672920, −4.41567397460630401639908946543, −3.66011707223585551842612929798, −1.22210892297016557366668286222, 1.12701429751422840745182061166, 1.65865002449760006242014662737, 3.08109821854097978778919265357, 3.73662778041994071822886444527, 4.78139097802342127876190024829, 5.89473151011349848740658270996, 6.99531857147751877610840139931, 8.306899388302958389205787877787, 8.645032118896556432811057142400, 9.634293098771160563379977626850

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