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Properties

Label 2-1110-185.174-c1-0-28
Degree $2$
Conductor $1110$
Sign $-0.206 + 0.978i$
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  + (−0.866 + 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s + (−1.62 + 1.53i)5-s − 0.999·6-s + (−0.149 − 0.0860i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (0.634 − 2.14i)10-s − 4.78·11-s + (0.866 − 0.499i)12-s + (−0.684 − 0.395i)13-s + 0.172·14-s + (−2.17 + 0.522i)15-s + (−0.5 − 0.866i)16-s + (1.23 − 0.711i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.499 + 0.288i)3-s + (0.249 − 0.433i)4-s + (−0.725 + 0.688i)5-s − 0.408·6-s + (−0.0563 − 0.0325i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (0.200 − 0.678i)10-s − 1.44·11-s + (0.249 − 0.144i)12-s + (−0.189 − 0.109i)13-s + 0.0460·14-s + (−0.561 + 0.135i)15-s + (−0.125 − 0.216i)16-s + (0.298 − 0.172i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.206 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.206 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.206 + 0.978i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (1099, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.206 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2328101042\)
\(L(\frac12)\) \(\approx\) \(0.2328101042\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (1.62 - 1.53i)T \)
37 \( 1 + (-4.44 + 4.14i)T \)
good7 \( 1 + (0.149 + 0.0860i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + 4.78T + 11T^{2} \)
13 \( 1 + (0.684 + 0.395i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.23 + 0.711i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.532 - 0.921i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 6.69iT - 23T^{2} \)
29 \( 1 + 10.2T + 29T^{2} \)
31 \( 1 - 4.87T + 31T^{2} \)
41 \( 1 + (2.43 - 4.21i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 5.11iT - 43T^{2} \)
47 \( 1 + 0.109iT - 47T^{2} \)
53 \( 1 + (-0.219 + 0.126i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.429 + 0.744i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.58 - 7.94i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (12.0 + 6.95i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.56 + 7.90i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 1.10iT - 73T^{2} \)
79 \( 1 + (1.50 - 2.61i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.06 - 1.19i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.78 - 3.09i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10.3iT - 97T^{2} \)
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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

−9.639590487521639268490757012304, −8.632599108008958914442505367656, −7.87496245586972043917109958429, −7.46760681448587918630581170185, −6.45206231831703875717123717244, −5.37404315494265900904484937291, −4.32850723784565565282046954572, −3.14027077059041606723156129161, −2.28261553254189713751156841061, −0.11578343246428925790506727796, 1.45041380781392758434189811372, 2.73016320456839176052656589460, 3.67456243179492352833654261245, 4.80551950103100016866497412762, 5.82130521507333258152795229102, 7.26001048168168911616146356860, 7.76606742815565937738338359778, 8.335402359242277954927172789968, 9.272173300576279754290703573395, 9.857044380527956918002461869442

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