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Properties

Label 2-1110-185.84-c1-0-37
Degree $2$
Conductor $1110$
Sign $-0.359 + 0.933i$
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 + (−0.522 − 2.17i)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 + (1.53 + 1.62i)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 + (−0.233 − 0.972i)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 + (0.397 + 0.418i)15-s + (−0.125 + 0.216i)16-s + (−0.298 − 0.172i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.359 + 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.359 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.359 + 0.933i$
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: \((2,\ 1110,\ (\ :1/2),\ -0.359 + 0.933i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7401174285\)
\(L(\frac12)\) \(\approx\) \(0.7401174285\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.522 + 2.17i)T \)
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

−9.552598494640037589276132401820, −8.618115441843053991343718042797, −7.925486599104919029060952533000, −7.03443810292343314088818091271, −5.92487030395823528958178078378, −5.19600648597850897951359937338, −4.61083746910929442433916032513, −3.60858151126409430271398496248, −2.21118064141460089978607026169, −0.26258071000966266469162256472, 1.81619841401263821600680338258, 2.90048590296535300242991486675, 3.83504281583583648834087214957, 5.05322338129766808666028317870, 5.77171762202637187275262676823, 6.65294720628615156559003439802, 7.48718706937889958097025922856, 8.175926186187051914165318224902, 9.650134071504853555521035718735, 10.31290355065061876672057771977

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