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Properties

Label 2-1155-77.76-c1-0-49
Degree $2$
Conductor $1155$
Sign $0.390 + 0.920i$
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  − 0.474i·2-s + i·3-s + 1.77·4-s − i·5-s + 0.474·6-s + (−0.474 − 2.60i)7-s − 1.79i·8-s − 9-s − 0.474·10-s + (3.23 − 0.726i)11-s + 1.77i·12-s + 0.814·13-s + (−1.23 + 0.225i)14-s + 15-s + 2.69·16-s − 6.40·17-s + ⋯
L(s)  = 1  − 0.335i·2-s + 0.577i·3-s + 0.887·4-s − 0.447i·5-s + 0.193·6-s + (−0.179 − 0.983i)7-s − 0.633i·8-s − 0.333·9-s − 0.150·10-s + (0.975 − 0.219i)11-s + 0.512i·12-s + 0.225·13-s + (−0.330 + 0.0602i)14-s + 0.258·15-s + 0.674·16-s − 1.55·17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.390 + 0.920i$
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: \((2,\ 1155,\ (\ :1/2),\ 0.390 + 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.999922587\)
\(L(\frac12)\) \(\approx\) \(1.999922587\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.474 + 2.60i)T \)
11 \( 1 + (-3.23 + 0.726i)T \)
good2 \( 1 + 0.474iT - 2T^{2} \)
13 \( 1 - 0.814T + 13T^{2} \)
17 \( 1 + 6.40T + 17T^{2} \)
19 \( 1 - 4.95T + 19T^{2} \)
23 \( 1 + 0.774T + 23T^{2} \)
29 \( 1 + 2.51iT - 29T^{2} \)
31 \( 1 + 5.42iT - 31T^{2} \)
37 \( 1 + 5.42T + 37T^{2} \)
41 \( 1 + 2.77T + 41T^{2} \)
43 \( 1 + 6.55iT - 43T^{2} \)
47 \( 1 - 4.57iT - 47T^{2} \)
53 \( 1 - 9.63T + 53T^{2} \)
59 \( 1 + 3.34iT - 59T^{2} \)
61 \( 1 - 8.03T + 61T^{2} \)
67 \( 1 - 16.0T + 67T^{2} \)
71 \( 1 + 3.04T + 71T^{2} \)
73 \( 1 - 9.23T + 73T^{2} \)
79 \( 1 - 7.97iT - 79T^{2} \)
83 \( 1 + 6.58T + 83T^{2} \)
89 \( 1 - 1.12iT - 89T^{2} \)
97 \( 1 - 12.2iT - 97T^{2} \)
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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

−9.755147830758410184838804770369, −9.031025552592475759196277604507, −8.025973582408236553739186414898, −7.00229962427981759470352220794, −6.44043303186897711477168278659, −5.32510303496649746124131817291, −4.08752073473350716074867863970, −3.61609009587292384034180586012, −2.22743377239064447345885648075, −0.885302855003492806782493718894, 1.62994805737110104805227979420, 2.53626196269337117016542394561, 3.52844768176755705134547445265, 5.11044849888520011109296148489, 5.99187168562588154209173624538, 6.79739608317458082295412383600, 7.07549259147004822942188717405, 8.325071421996745808013853464081, 8.902431421851688400810153073796, 9.907638237933511599590513139931

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