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Properties

Label 2-1110-37.26-c1-0-9
Degree $2$
Conductor $1110$
Sign $0.994 - 0.107i$
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 + 11-s + (0.499 + 0.866i)12-s + (−0.5 + 0.866i)13-s + (0.499 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (3.88 + 6.73i)17-s + (−0.499 + 0.866i)18-s + (−2.38 + 4.13i)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 + 0.301·11-s + (0.144 + 0.249i)12-s + (−0.138 + 0.240i)13-s + (0.129 + 0.223i)15-s + (−0.125 − 0.216i)16-s + (0.942 + 1.63i)17-s + (−0.117 + 0.204i)18-s + (−0.547 + 0.948i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.107i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.994 - 0.107i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.994 - 0.107i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.266499650\)
\(L(\frac12)\) \(\approx\) \(1.266499650\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.27 + 4.33i)T \)
good7 \( 1 + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - T + 11T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.88 - 6.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.38 - 4.13i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 2.77T + 23T^{2} \)
29 \( 1 - 0.227T + 29T^{2} \)
31 \( 1 - 9.54T + 31T^{2} \)
41 \( 1 + (2.77 - 4.80i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 3T + 47T^{2} \)
53 \( 1 + (-6.77 - 11.7i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.27 - 10.8i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.65 + 13.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.77 - 3.06i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + (2.77 - 4.80i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.77 + 8.26i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.61 - 6.25i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 0.455T + 97T^{2} \)
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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

−10.11974573759229800255355867717, −8.963272043059590078085836820019, −8.200458178329801329932942226368, −7.66402517944184679628904814496, −6.55541157616001055584417837006, −5.80955872350483288171899526094, −4.24034892784147901542605763757, −3.53159634148570540233068994289, −2.35950022580892239377092249392, −1.28961049983078420308403496095, 0.68480709441086019028028159324, 2.50187744033073840836065796300, 3.74960993764071336070541131439, 4.83096513932893345102126868925, 5.39770625872568395989459557434, 6.65898374882484104006679926809, 7.37359904241933993978707337000, 8.381124219962166606505774349531, 8.817429420237618194332331841231, 9.887674655456936014578300190487

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