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Properties

Label 2-888-24.11-c1-0-49
Degree $2$
Conductor $888$
Sign $-0.707 - 0.707i$
Analytic cond. $7.09071$
Root an. cond. $2.66283$
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  + (1.36 + 0.366i)2-s + 1.73i·3-s + (1.73 + i)4-s − 0.732·5-s + (−0.633 + 2.36i)6-s + 2i·7-s + (1.99 + 2i)8-s − 2.99·9-s + (−1 − 0.267i)10-s − 1.46i·11-s + (−1.73 + 2.99i)12-s + 2i·13-s + (−0.732 + 2.73i)14-s − 1.26i·15-s + (1.99 + 3.46i)16-s + 2.73i·17-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + 0.999i·3-s + (0.866 + 0.5i)4-s − 0.327·5-s + (−0.258 + 0.965i)6-s + 0.755i·7-s + (0.707 + 0.707i)8-s − 0.999·9-s + (−0.316 − 0.0847i)10-s − 0.441i·11-s + (−0.499 + 0.866i)12-s + 0.554i·13-s + (−0.195 + 0.730i)14-s − 0.327i·15-s + (0.499 + 0.866i)16-s + 0.662i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(7.09071\)
Root analytic conductor: \(2.66283\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{888} (371, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 888,\ (\ :1/2),\ -0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.944660 + 2.28061i\)
\(L(\frac12)\) \(\approx\) \(0.944660 + 2.28061i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 - 0.366i)T \)
3 \( 1 - 1.73iT \)
37 \( 1 + iT \)
good5 \( 1 + 0.732T + 5T^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + 1.46iT - 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 2.73iT - 17T^{2} \)
19 \( 1 + 6.19T + 19T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 - 10.1T + 29T^{2} \)
31 \( 1 + 4.19iT - 31T^{2} \)
41 \( 1 - 9.46iT - 41T^{2} \)
43 \( 1 + 3.26T + 43T^{2} \)
47 \( 1 + 3.46T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 + 3.46iT - 59T^{2} \)
61 \( 1 + 8.53iT - 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 15.4T + 71T^{2} \)
73 \( 1 - 4.92T + 73T^{2} \)
79 \( 1 - 11.1iT - 79T^{2} \)
83 \( 1 + 14.3iT - 83T^{2} \)
89 \( 1 - 9.66iT - 89T^{2} \)
97 \( 1 - 15.8T + 97T^{2} \)
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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

−10.70439401675423559521715031123, −9.643387445499787599640287203868, −8.544801931131048969226560193481, −8.144221248496717364628561072880, −6.61019968104621794972729847005, −6.03253490690761630253721717637, −4.98359865847112074205689893567, −4.28245655431936700685705876208, −3.33696440904659457335346902161, −2.30820956729624527971094338171, 0.861773648917964766796216843560, 2.25342949529353814261914331170, 3.28400065902590512228719706150, 4.43134923400772356077887098080, 5.34247562530831811079202101263, 6.51015518331160171089781660893, 7.02192554855406089162372336623, 7.81713162337979593933995264343, 8.812344071384776966002507690869, 10.23197256196749556258932972611

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