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Properties

Label 2-1110-37.26-c1-0-20
Degree $2$
Conductor $1110$
Sign $0.729 + 0.683i$
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  + (0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + 0.999·6-s − 0.999·8-s + (−0.499 − 0.866i)9-s + 0.999·10-s − 3·11-s + (0.499 + 0.866i)12-s + (2.5 − 4.33i)13-s + (−0.499 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−1 − 1.73i)17-s + (0.499 − 0.866i)18-s + (2.5 − 4.33i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.223 − 0.387i)5-s + 0.408·6-s − 0.353·8-s + (−0.166 − 0.288i)9-s + 0.316·10-s − 0.904·11-s + (0.144 + 0.249i)12-s + (0.693 − 1.20i)13-s + (−0.129 − 0.223i)15-s + (−0.125 − 0.216i)16-s + (−0.242 − 0.420i)17-s + (0.117 − 0.204i)18-s + (0.573 − 0.993i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 + 0.683i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.729 + 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.729 + 0.683i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.729 + 0.683i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.955179215\)
\(L(\frac12)\) \(\approx\) \(1.955179215\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.5 + 6.06i)T \)
good7 \( 1 + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + (-2.5 + 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 5T + 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
41 \( 1 + (1 - 1.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + 3T + 47T^{2} \)
53 \( 1 + (1 + 1.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.5 + 9.52i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3 + 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1 + 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2 - 3.46i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 + (-1 + 1.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7 - 12.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.5 - 4.33i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 16T + 97T^{2} \)
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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

−9.451527764073938227669520685484, −8.778575515497031681308082004788, −7.952092881369026163692617986427, −7.33284778826714492784211051714, −6.38661684263454426976783781788, −5.43264075326392101073570722249, −4.84736269019168766151916501055, −3.40622043192681713601759837762, −2.54712041799946166243148096999, −0.77068668204527063927287674008, 1.60514424115448976317171268497, 2.76778029490586732208383663897, 3.65289029493915044012674541124, 4.58560061388893452527437944368, 5.52512989908406569372306227121, 6.43070490361064058370796256610, 7.48300381395437310381146618550, 8.575555120512153594599415289318, 9.206007835969312755106643406564, 10.18292189491926584971762581825

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