Properties

Label 2-1110-185.84-c1-0-39
Degree $2$
Conductor $1110$
Sign $-0.955 + 0.293i$
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 + (−0.329 − 2.21i)5-s − 0.999·6-s + (0.903 − 0.521i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (−0.820 + 2.07i)10-s + 0.877·11-s + (0.866 + 0.499i)12-s + (−3.82 + 2.20i)13-s − 1.04·14-s + (−1.39 − 1.75i)15-s + (−0.5 + 0.866i)16-s + (−4.44 − 2.56i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.147 − 0.989i)5-s − 0.408·6-s + (0.341 − 0.197i)7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.259 + 0.657i)10-s + 0.264·11-s + (0.249 + 0.144i)12-s + (−1.05 + 0.611i)13-s − 0.278·14-s + (−0.359 − 0.452i)15-s + (−0.125 + 0.216i)16-s + (−1.07 − 0.622i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 + 0.293i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.955 + 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.955 + 0.293i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.955 + 0.293i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9336770071\)
\(L(\frac12)\) \(\approx\) \(0.9336770071\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.329 + 2.21i)T \)
37 \( 1 + (0.0184 + 6.08i)T \)
good7 \( 1 + (-0.903 + 0.521i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 - 0.877T + 11T^{2} \)
13 \( 1 + (3.82 - 2.20i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (4.44 + 2.56i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.51 + 2.62i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 7.11iT - 23T^{2} \)
29 \( 1 - 5.96T + 29T^{2} \)
31 \( 1 + 1.73T + 31T^{2} \)
41 \( 1 + (-3.12 - 5.41i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 1.46iT - 43T^{2} \)
47 \( 1 - 3.00iT - 47T^{2} \)
53 \( 1 + (11.1 + 6.44i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.32 + 7.48i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.80 - 3.13i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.30 - 1.33i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.582 - 1.00i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 5.73iT - 73T^{2} \)
79 \( 1 + (5.03 + 8.72i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.72 + 5.61i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.63 + 13.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.07iT - 97T^{2} \)
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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

−9.258376662069459396232561713880, −8.787000814805981825548692003361, −8.027326112114673564753732451924, −7.17603667334308222448040994496, −6.38862346778881905346488995505, −4.72878667748577373977394929940, −4.37088121021911010221518075392, −2.77108612191599214041689570870, −1.84376136897462936538235863911, −0.44939514485174928491940230701, 1.86889664233860349970425823284, 2.87368438282599510159961857195, 3.99689361384490923505596860934, 5.16395683587291806235518460076, 6.21512989013487633148238684557, 7.06516105464813699407908618927, 7.81393851557972334240083944572, 8.471711998824937554875240387854, 9.433981741552631582270227306613, 10.13033839438143918584146605593

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