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Properties

Label 2-888-296.179-c1-0-68
Degree $2$
Conductor $888$
Sign $0.0824 - 0.996i$
Analytic cond. $7.09071$
Root an. cond. $2.66283$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − i)2-s − i·3-s + 2i·4-s + (−2 − 2i)5-s + (−1 + i)6-s + 2i·7-s + (2 − 2i)8-s − 9-s + 4i·10-s − 4i·11-s + 2·12-s + (−1 − i)13-s + (2 − 2i)14-s + (−2 + 2i)15-s − 4·16-s + (−2 + 2i)17-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s − 0.577i·3-s + i·4-s + (−0.894 − 0.894i)5-s + (−0.408 + 0.408i)6-s + 0.755i·7-s + (0.707 − 0.707i)8-s − 0.333·9-s + 1.26i·10-s − 1.20i·11-s + 0.577·12-s + (−0.277 − 0.277i)13-s + (0.534 − 0.534i)14-s + (−0.516 + 0.516i)15-s − 16-s + (−0.485 + 0.485i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0824 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0824 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $0.0824 - 0.996i$
Analytic conductor: \(7.09071\)
Root analytic conductor: \(2.66283\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{888} (475, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 888,\ (\ :1/2),\ 0.0824 - 0.996i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
3 \( 1 + iT \)
37 \( 1 + (1 - 6i)T \)
good5 \( 1 + (2 + 2i)T + 5iT^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + 4iT - 11T^{2} \)
13 \( 1 + (1 + i)T + 13iT^{2} \)
17 \( 1 + (2 - 2i)T - 17iT^{2} \)
19 \( 1 + (-1 + i)T - 19iT^{2} \)
23 \( 1 + (2 + 2i)T + 23iT^{2} \)
29 \( 1 + (2 - 2i)T - 29iT^{2} \)
31 \( 1 + (7 - 7i)T - 31iT^{2} \)
41 \( 1 - 2iT - 41T^{2} \)
43 \( 1 + (-3 + 3i)T - 43iT^{2} \)
47 \( 1 - 4iT - 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 + (8 - 8i)T - 59iT^{2} \)
61 \( 1 + (1 - i)T - 61iT^{2} \)
67 \( 1 - 2iT - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 + (3 + 3i)T + 79iT^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 89iT^{2} \)
97 \( 1 + (-3 + 3i)T - 97iT^{2} \)
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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

−9.108133118196141956323796647472, −8.720641910488239256617846959335, −8.099025046418269072891519114432, −7.27314193888750601323625848807, −6.06767403421371306206175065841, −4.91607225912987517143439621329, −3.73035769981333388573314648440, −2.72766549364153072777069575179, −1.33002203539019968278374339651, 0, 2.15333976409699628574921211347, 3.78090169944807331430310030282, 4.50867558539335250968776920496, 5.66648279770162576168985679613, 6.91534628999212558492856793582, 7.34847979860524319173576549594, 7.981126261820416233182584945004, 9.321924135994664025758133813425, 9.738633219224020670869408153345

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