Properties

Label 2-1155-77.76-c1-0-15
Degree $2$
Conductor $1155$
Sign $0.299 + 0.954i$
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  − 2.52i·2-s i·3-s − 4.37·4-s + i·5-s − 2.52·6-s + (−2.52 + 0.792i)7-s + 5.98i·8-s − 9-s + 2.52·10-s + 3.31i·11-s + 4.37i·12-s + (2 + 6.37i)14-s + 15-s + 6.37·16-s + 5.84·17-s + 2.52i·18-s + ⋯
L(s)  = 1  − 1.78i·2-s − 0.577i·3-s − 2.18·4-s + 0.447i·5-s − 1.03·6-s + (−0.954 + 0.299i)7-s + 2.11i·8-s − 0.333·9-s + 0.798·10-s + 1.00i·11-s + 1.26i·12-s + (0.534 + 1.70i)14-s + 0.258·15-s + 1.59·16-s + 1.41·17-s + 0.594i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.299 + 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.299 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.299 + 0.954i$
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: \((2,\ 1155,\ (\ :1/2),\ 0.299 + 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.051320197\)
\(L(\frac12)\) \(\approx\) \(1.051320197\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.52 - 0.792i)T \)
11 \( 1 - 3.31iT \)
good2 \( 1 + 2.52iT - 2T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 5.84T + 17T^{2} \)
19 \( 1 - 7.72T + 19T^{2} \)
23 \( 1 + 8.11T + 23T^{2} \)
29 \( 1 + 2.67iT - 29T^{2} \)
31 \( 1 - 4.74iT - 31T^{2} \)
37 \( 1 + 0.744T + 37T^{2} \)
41 \( 1 - 8.21T + 41T^{2} \)
43 \( 1 - 5.84iT - 43T^{2} \)
47 \( 1 - 1.25iT - 47T^{2} \)
53 \( 1 + 7.37T + 53T^{2} \)
59 \( 1 - 7.37iT - 59T^{2} \)
61 \( 1 - 1.08T + 61T^{2} \)
67 \( 1 - 6.74T + 67T^{2} \)
71 \( 1 - 14.7T + 71T^{2} \)
73 \( 1 + 6.92T + 73T^{2} \)
79 \( 1 - 13.5iT - 79T^{2} \)
83 \( 1 - 6.13T + 83T^{2} \)
89 \( 1 + 2.62iT - 89T^{2} \)
97 \( 1 - 2.62iT - 97T^{2} \)
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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

−9.746015714047859343912903877588, −9.416953527467515200714502434306, −8.082503736123891681085428945598, −7.32719652201902499644873425577, −6.13675409047281552688735134307, −5.18494517302661382990757746898, −3.87909451358319006249920371612, −3.11622821320990349840862951742, −2.30116091884957176549286589863, −1.11249727019631978645235681516, 0.56307888095912163101929210505, 3.31940819259457720067691578601, 4.01212513720948654111131835193, 5.28066215601559832149499592079, 5.72687409575920514242487424201, 6.47915611700006308796209857464, 7.66057642597554793988525470598, 8.001695299911443631053511236981, 9.121382838801864580489104043488, 9.586257702696309922203597997674

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