Properties

 Label 2-1110-185.68-c1-0-6 Degree $2$ Conductor $1110$ Sign $-0.787 - 0.615i$ Analytic cond. $8.86339$ Root an. cond. $2.97714$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 − 2-s + (−0.707 + 0.707i)3-s + 4-s + (−0.866 + 2.06i)5-s + (0.707 − 0.707i)6-s + (−0.662 + 0.662i)7-s − 8-s − 1.00i·9-s + (0.866 − 2.06i)10-s + 3.51i·11-s + (−0.707 + 0.707i)12-s + 6.87·13-s + (0.662 − 0.662i)14-s + (−0.844 − 2.07i)15-s + 16-s + 2.17i·17-s + ⋯
 L(s)  = 1 − 0.707·2-s + (−0.408 + 0.408i)3-s + 0.5·4-s + (−0.387 + 0.921i)5-s + (0.288 − 0.288i)6-s + (−0.250 + 0.250i)7-s − 0.353·8-s − 0.333i·9-s + (0.273 − 0.651i)10-s + 1.06i·11-s + (−0.204 + 0.204i)12-s + 1.90·13-s + (0.176 − 0.176i)14-s + (−0.218 − 0.534i)15-s + 0.250·16-s + 0.527i·17-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.787 - 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.787 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$1110$$    =    $$2 \cdot 3 \cdot 5 \cdot 37$$ Sign: $-0.787 - 0.615i$ Analytic conductor: $$8.86339$$ Root analytic conductor: $$2.97714$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1110} (253, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1110,\ (\ :1/2),\ -0.787 - 0.615i)$$

Particular Values

 $$L(1)$$ $$\approx$$ $$0.7966429543$$ $$L(\frac12)$$ $$\approx$$ $$0.7966429543$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + T$$
3 $$1 + (0.707 - 0.707i)T$$
5 $$1 + (0.866 - 2.06i)T$$
37 $$1 + (5.95 - 1.24i)T$$
good7 $$1 + (0.662 - 0.662i)T - 7iT^{2}$$
11 $$1 - 3.51iT - 11T^{2}$$
13 $$1 - 6.87T + 13T^{2}$$
17 $$1 - 2.17iT - 17T^{2}$$
19 $$1 + (1.01 + 1.01i)T + 19iT^{2}$$
23 $$1 - 9.57T + 23T^{2}$$
29 $$1 + (6.49 - 6.49i)T - 29iT^{2}$$
31 $$1 + (-2.38 - 2.38i)T + 31iT^{2}$$
41 $$1 - 3.47iT - 41T^{2}$$
43 $$1 - 2.74T + 43T^{2}$$
47 $$1 + (-3.59 + 3.59i)T - 47iT^{2}$$
53 $$1 + (2.93 + 2.93i)T + 53iT^{2}$$
59 $$1 + (3.45 + 3.45i)T + 59iT^{2}$$
61 $$1 + (9.99 + 9.99i)T + 61iT^{2}$$
67 $$1 + (1.14 + 1.14i)T + 67iT^{2}$$
71 $$1 + 7.32T + 71T^{2}$$
73 $$1 + (0.292 - 0.292i)T - 73iT^{2}$$
79 $$1 + (-7.95 - 7.95i)T + 79iT^{2}$$
83 $$1 + (1.46 + 1.46i)T + 83iT^{2}$$
89 $$1 + (6.43 - 6.43i)T - 89iT^{2}$$
97 $$1 + 0.301iT - 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

−10.34253896194134439669895884927, −9.275928076680671557750010479662, −8.739880829219058099761221217765, −7.65264680290486700310149646162, −6.78014817075995029508937948417, −6.27455760548615363351672168655, −5.09609641175702767481643015589, −3.81458627450702843029826680902, −3.02444537626859866075993130877, −1.49135154614572352778357341868, 0.53200132611209224297353088596, 1.41126939299795625911107540055, 3.15733717831205014314321182481, 4.17639537161670799610443590109, 5.54105243347720167507378065258, 6.11218647693178956521246960714, 7.16906499669907977292116622905, 7.972603230542320031698178807603, 8.829960540852821145947816256353, 9.134728569106580504582438256716