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Properties

Label 2-1110-185.174-c1-0-37
Degree $2$
Conductor $1110$
Sign $-0.982 + 0.184i$
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 + (−0.221 − 2.22i)5-s − 0.999·6-s + (−0.666 − 0.384i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (1.30 + 1.81i)10-s − 0.542·11-s + (0.866 − 0.499i)12-s + (−5.71 − 3.29i)13-s + 0.769·14-s + (0.921 − 2.03i)15-s + (−0.5 − 0.866i)16-s + (−3.00 + 1.73i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.499 + 0.288i)3-s + (0.249 − 0.433i)4-s + (−0.0988 − 0.995i)5-s − 0.408·6-s + (−0.251 − 0.145i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (0.412 + 0.574i)10-s − 0.163·11-s + (0.249 − 0.144i)12-s + (−1.58 − 0.914i)13-s + 0.205·14-s + (0.237 − 0.526i)15-s + (−0.125 − 0.216i)16-s + (−0.728 + 0.420i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.184i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.982 + 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.982 + 0.184i$
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: \((2,\ 1110,\ (\ :1/2),\ -0.982 + 0.184i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.08015711344\)
\(L(\frac12)\) \(\approx\) \(0.08015711344\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.221 + 2.22i)T \)
37 \( 1 + (1.40 + 5.91i)T \)
good7 \( 1 + (0.666 + 0.384i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + 0.542T + 11T^{2} \)
13 \( 1 + (5.71 + 3.29i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.00 - 1.73i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.29 - 2.23i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 3.05iT - 23T^{2} \)
29 \( 1 - 1.93T + 29T^{2} \)
31 \( 1 + 2.81T + 31T^{2} \)
41 \( 1 + (4.40 - 7.63i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 6.99iT - 43T^{2} \)
47 \( 1 - 11.5iT - 47T^{2} \)
53 \( 1 + (-3.51 + 2.02i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.29 + 3.96i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.02 - 3.50i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.50 + 4.91i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.34 - 2.33i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 9.10iT - 73T^{2} \)
79 \( 1 + (-2.97 + 5.15i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.26 + 4.77i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.24 + 10.8i)T + (-44.5 + 77.0i)T^{2} \)
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

−9.490484897313792854819481313095, −8.603370468832673012261422424610, −7.918173556369146723814339592204, −7.30089960798287294715268411667, −6.05968550019886304980709989555, −5.11973567675971511490929189142, −4.34433304505674586483712022208, −2.99838034834055868961598670827, −1.72547511319854253917721993221, −0.03703641648361360838220051773, 2.14909756863139796366530376622, 2.63575180656627147105780713474, 3.80206178992416086009872380738, 4.95961081548309438356960168746, 6.55727567383949504971994181901, 6.97106319630602983025388960458, 7.69181179386615330786005474915, 8.764898502330880938020996380730, 9.364147059600030504310154593218, 10.22090103819777104736793591843

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