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Properties

Label 2-888-111.14-c1-0-33
Degree $2$
Conductor $888$
Sign $-0.974 - 0.224i$
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  + (−0.728 − 1.57i)3-s + (0.601 − 2.24i)5-s + (−0.786 − 1.36i)7-s + (−1.93 + 2.28i)9-s + 0.738·11-s + (−5.22 − 1.40i)13-s + (−3.96 + 0.689i)15-s + (6.88 − 1.84i)17-s + (−2.79 − 0.749i)19-s + (−1.56 + 2.22i)21-s + (−5.90 − 5.90i)23-s + (−0.345 − 0.199i)25-s + (5.00 + 1.38i)27-s + (−5.65 + 5.65i)29-s + (−1.19 − 1.19i)31-s + ⋯
L(s)  = 1  + (−0.420 − 0.907i)3-s + (0.268 − 1.00i)5-s + (−0.297 − 0.514i)7-s + (−0.646 + 0.762i)9-s + 0.222·11-s + (−1.45 − 0.388i)13-s + (−1.02 + 0.177i)15-s + (1.66 − 0.447i)17-s + (−0.641 − 0.172i)19-s + (−0.342 + 0.486i)21-s + (−1.23 − 1.23i)23-s + (−0.0691 − 0.0399i)25-s + (0.963 + 0.265i)27-s + (−1.05 + 1.05i)29-s + (−0.215 − 0.215i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 - 0.224i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 - 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $-0.974 - 0.224i$
Analytic conductor: \(7.09071\)
Root analytic conductor: \(2.66283\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{888} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 888,\ (\ :1/2),\ -0.974 - 0.224i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0847374 + 0.744475i\)
\(L(\frac12)\) \(\approx\) \(0.0847374 + 0.744475i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.728 + 1.57i)T \)
37 \( 1 + (-4.55 - 4.03i)T \)
good5 \( 1 + (-0.601 + 2.24i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (0.786 + 1.36i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 0.738T + 11T^{2} \)
13 \( 1 + (5.22 + 1.40i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-6.88 + 1.84i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (2.79 + 0.749i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (5.90 + 5.90i)T + 23iT^{2} \)
29 \( 1 + (5.65 - 5.65i)T - 29iT^{2} \)
31 \( 1 + (1.19 + 1.19i)T + 31iT^{2} \)
41 \( 1 + (-4.27 - 7.41i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (7.64 - 7.64i)T - 43iT^{2} \)
47 \( 1 + 10.6iT - 47T^{2} \)
53 \( 1 + (4.00 + 2.30i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.81 + 0.755i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.69 - 10.0i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (5.21 - 3.00i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.51 - 1.44i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 14.9iT - 73T^{2} \)
79 \( 1 + (1.59 + 0.426i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (12.2 + 7.06i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.62 + 13.5i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-0.824 + 0.824i)T - 97iT^{2} \)
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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

−9.824618512309704572796443823672, −8.715506329360365821486555246006, −7.85499909461341326228185534958, −7.20601658045880549985390361653, −6.18511482049050190908993912913, −5.30608983702374608205755378467, −4.54756406800897038777836303058, −2.96327208718495648404887830549, −1.62055977909699153196382461585, −0.36636744227454104219497583009, 2.22819485068870758268208538489, 3.33488705895428297245729977163, 4.22991078167244762374487789020, 5.59225922860123326943086793866, 5.96418290426131476519908200048, 7.11189446546954508171771127822, 7.988864863465389241538138742785, 9.375352959650433177658196258381, 9.739119648050017514664738990299, 10.42823614128708255039244786081

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