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Properties

Label 2-1110-5.4-c1-0-0
Degree $2$
Conductor $1110$
Sign $0.158 + 0.987i$
Analytic cond. $8.86339$
Root an. cond. $2.97714$
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  + i·2-s + i·3-s − 4-s + (−2.20 + 0.353i)5-s − 6-s + 4.51i·7-s − i·8-s − 9-s + (−0.353 − 2.20i)10-s − 6.51·11-s − i·12-s + 0.854i·13-s − 4.51·14-s + (−0.353 − 2.20i)15-s + 16-s + 1.38i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (−0.987 + 0.158i)5-s − 0.408·6-s + 1.70i·7-s − 0.353i·8-s − 0.333·9-s + (−0.111 − 0.698i)10-s − 1.96·11-s − 0.288i·12-s + 0.236i·13-s − 1.20·14-s + (−0.0913 − 0.570i)15-s + 0.250·16-s + 0.336i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.158 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.158 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.158 + 0.987i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.158 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2735559242\)
\(L(\frac12)\) \(\approx\) \(0.2735559242\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 - iT \)
5 \( 1 + (2.20 - 0.353i)T \)
37 \( 1 + iT \)
good7 \( 1 - 4.51iT - 7T^{2} \)
11 \( 1 + 6.51T + 11T^{2} \)
13 \( 1 - 0.854iT - 13T^{2} \)
17 \( 1 - 1.38iT - 17T^{2} \)
19 \( 1 - 7.74T + 19T^{2} \)
23 \( 1 + 4.68iT - 23T^{2} \)
29 \( 1 - 2.36T + 29T^{2} \)
31 \( 1 + 9.17T + 31T^{2} \)
41 \( 1 - 4.17T + 41T^{2} \)
43 \( 1 - 3.65iT - 43T^{2} \)
47 \( 1 + 10.4iT - 47T^{2} \)
53 \( 1 - 9.61iT - 53T^{2} \)
59 \( 1 + 6.41T + 59T^{2} \)
61 \( 1 + 10.9T + 61T^{2} \)
67 \( 1 + 12.9iT - 67T^{2} \)
71 \( 1 + 0.0391T + 71T^{2} \)
73 \( 1 - 7.16iT - 73T^{2} \)
79 \( 1 - 7.35T + 79T^{2} \)
83 \( 1 + 10.9iT - 83T^{2} \)
89 \( 1 + 5.73T + 89T^{2} \)
97 \( 1 - 11.6iT - 97T^{2} \)
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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

−10.46606650921639800661342765337, −9.405990441502810559918229821295, −8.758969041936060871527953403161, −7.973293237156583831180840593544, −7.41901465674559593172013919159, −6.08345090118077958166490020447, −5.31042560627243182463673533001, −4.77017698114219158840595286940, −3.37463231593500615219146090466, −2.55819643038665427589793077870, 0.13339224413338955962918997887, 1.21237197701395780407125795517, 2.92429757324220568683528041095, 3.61650579244009096499461393252, 4.75372919326764564835098399688, 5.50377312681547330173654112243, 7.22833482689677307640895054134, 7.55519370687951617108333831048, 8.067853852509170721408029043506, 9.368781616484116211254508559517

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