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Properties

Label 2-1155-5.4-c1-0-26
Degree $2$
Conductor $1155$
Sign $0.330 - 0.943i$
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  + 2.74i·2-s − i·3-s − 5.51·4-s + (−0.739 + 2.11i)5-s + 2.74·6-s − i·7-s − 9.63i·8-s − 9-s + (−5.78 − 2.02i)10-s + 11-s + 5.51i·12-s − 3.27i·13-s + 2.74·14-s + (2.11 + 0.739i)15-s + 15.3·16-s + 0.321i·17-s + ⋯
L(s)  = 1  + 1.93i·2-s − 0.577i·3-s − 2.75·4-s + (−0.330 + 0.943i)5-s + 1.11·6-s − 0.377i·7-s − 3.40i·8-s − 0.333·9-s + (−1.82 − 0.641i)10-s + 0.301·11-s + 1.59i·12-s − 0.908i·13-s + 0.732·14-s + (0.544 + 0.190i)15-s + 3.84·16-s + 0.0778i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.330 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.330 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.330 - 0.943i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (694, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ 0.330 - 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.081934381\)
\(L(\frac12)\) \(\approx\) \(1.081934381\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.739 - 2.11i)T \)
7 \( 1 + iT \)
11 \( 1 - T \)
good2 \( 1 - 2.74iT - 2T^{2} \)
13 \( 1 + 3.27iT - 13T^{2} \)
17 \( 1 - 0.321iT - 17T^{2} \)
19 \( 1 - 3.74T + 19T^{2} \)
23 \( 1 + 8.57iT - 23T^{2} \)
29 \( 1 - 8.80T + 29T^{2} \)
31 \( 1 + 7.81T + 31T^{2} \)
37 \( 1 - 4.37iT - 37T^{2} \)
41 \( 1 - 2.60T + 41T^{2} \)
43 \( 1 - 7.99iT - 43T^{2} \)
47 \( 1 + 0.447iT - 47T^{2} \)
53 \( 1 + 11.0iT - 53T^{2} \)
59 \( 1 - 13.8T + 59T^{2} \)
61 \( 1 - 11.9T + 61T^{2} \)
67 \( 1 - 6.38iT - 67T^{2} \)
71 \( 1 + 7.35T + 71T^{2} \)
73 \( 1 + 5.14iT - 73T^{2} \)
79 \( 1 + 4.71T + 79T^{2} \)
83 \( 1 - 5.27iT - 83T^{2} \)
89 \( 1 - 8.21T + 89T^{2} \)
97 \( 1 + 16.8iT - 97T^{2} \)
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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

−9.813656568354105610357028229652, −8.553080602387585918921615385920, −8.125487566897254837241021396597, −7.26318031182639309262539036732, −6.76809183820387925602976842584, −6.10351587047222414619093220035, −5.14904246054140875900654752261, −4.11129428406098248836390772653, −3.03315709140841187248012346405, −0.62058260918966934916687669939, 1.05836571504063434266850934673, 2.17428391836219787255727210548, 3.48518956498780116620897070320, 4.05409529984009113925550257022, 5.00335277664457261443467146384, 5.60413391066275787158803619856, 7.53011436022991514467011515648, 8.660341937146707818737924562627, 9.128744684707230264324584958246, 9.622781759465411165963310265558

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