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Properties

Label 2-88-8.5-c5-0-14
Degree $2$
Conductor $88$
Sign $0.818 + 0.574i$
Analytic cond. $14.1137$
Root an. cond. $3.75683$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.22 − 3.76i)2-s − 2.50i·3-s + (3.69 + 31.7i)4-s − 0.695i·5-s + (−9.43 + 10.5i)6-s − 230.·7-s + (103. − 148. i)8-s + 236.·9-s + (−2.61 + 2.93i)10-s − 121i·11-s + (79.7 − 9.27i)12-s + 867. i·13-s + (971. + 865. i)14-s − 1.74·15-s + (−996. + 234. i)16-s + 1.09e3·17-s + ⋯
L(s)  = 1  + (−0.746 − 0.665i)2-s − 0.160i·3-s + (0.115 + 0.993i)4-s − 0.0124i·5-s + (−0.107 + 0.120i)6-s − 1.77·7-s + (0.574 − 0.818i)8-s + 0.974·9-s + (−0.00827 + 0.00929i)10-s − 0.301i·11-s + (0.159 − 0.0185i)12-s + 1.42i·13-s + (1.32 + 1.18i)14-s − 0.00200·15-s + (−0.973 + 0.229i)16-s + 0.921·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.818 + 0.574i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.818 + 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(88\)    =    \(2^{3} \cdot 11\)
Sign: $0.818 + 0.574i$
Analytic conductor: \(14.1137\)
Root analytic conductor: \(3.75683\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{88} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 88,\ (\ :5/2),\ 0.818 + 0.574i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.00248 - 0.316569i\)
\(L(\frac12)\) \(\approx\) \(1.00248 - 0.316569i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (4.22 + 3.76i)T \)
11 \( 1 + 121iT \)
good3 \( 1 + 2.50iT - 243T^{2} \)
5 \( 1 + 0.695iT - 3.12e3T^{2} \)
7 \( 1 + 230.T + 1.68e4T^{2} \)
13 \( 1 - 867. iT - 3.71e5T^{2} \)
17 \( 1 - 1.09e3T + 1.41e6T^{2} \)
19 \( 1 + 868. iT - 2.47e6T^{2} \)
23 \( 1 - 3.07e3T + 6.43e6T^{2} \)
29 \( 1 + 5.58e3iT - 2.05e7T^{2} \)
31 \( 1 - 1.91e3T + 2.86e7T^{2} \)
37 \( 1 + 6.58e3iT - 6.93e7T^{2} \)
41 \( 1 - 2.11e3T + 1.15e8T^{2} \)
43 \( 1 - 7.75e3iT - 1.47e8T^{2} \)
47 \( 1 - 1.79e4T + 2.29e8T^{2} \)
53 \( 1 - 3.87e4iT - 4.18e8T^{2} \)
59 \( 1 + 2.85e4iT - 7.14e8T^{2} \)
61 \( 1 - 3.55e4iT - 8.44e8T^{2} \)
67 \( 1 + 1.07e4iT - 1.35e9T^{2} \)
71 \( 1 + 2.04e4T + 1.80e9T^{2} \)
73 \( 1 - 1.61e4T + 2.07e9T^{2} \)
79 \( 1 - 3.48e4T + 3.07e9T^{2} \)
83 \( 1 - 1.14e5iT - 3.93e9T^{2} \)
89 \( 1 + 5.10e4T + 5.58e9T^{2} \)
97 \( 1 - 1.32e5T + 8.58e9T^{2} \)
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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

−12.85709913104107096737000755318, −12.09469373537639974959614606946, −10.75072149783801169363251622698, −9.654987244274027049585177895720, −9.077480180062638575055264700156, −7.33006867516596344361519376747, −6.50927531793608167543989928611, −4.10190117487592074805750784532, −2.76330008646574734302514937128, −0.884400561443663705082894448730, 0.847143800185374375267309700611, 3.22165593730616853590037215135, 5.26222711371858374213554799460, 6.57314640802918084477927557172, 7.46760012386789890863595150300, 8.951399858546911617978880611651, 10.03998229404244720602566481347, 10.44824285932248795778595924454, 12.49354772802847163336310017494, 13.19024601179981115799604535728

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