Properties

Label 2-88-11.3-c5-0-5
Degree $2$
Conductor $88$
Sign $-0.720 - 0.693i$
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  + (6.70 + 20.6i)3-s + (67.3 + 48.9i)5-s + (−74.4 + 229. i)7-s + (−184. + 134. i)9-s + (−79.9 − 393. i)11-s + (651. − 473. i)13-s + (−558. + 1.71e3i)15-s + (1.02e3 + 745. i)17-s + (−564. − 1.73e3i)19-s − 5.22e3·21-s − 3.37e3·23-s + (1.17e3 + 3.62e3i)25-s + (257. + 186. i)27-s + (−302. + 931. i)29-s + (−1.70e3 + 1.24e3i)31-s + ⋯
L(s)  = 1  + (0.430 + 1.32i)3-s + (1.20 + 0.875i)5-s + (−0.574 + 1.76i)7-s + (−0.760 + 0.552i)9-s + (−0.199 − 0.979i)11-s + (1.06 − 0.776i)13-s + (−0.641 + 1.97i)15-s + (0.861 + 0.626i)17-s + (−0.358 − 1.10i)19-s − 2.58·21-s − 1.32·23-s + (0.377 + 1.16i)25-s + (0.0679 + 0.0493i)27-s + (−0.0668 + 0.205i)29-s + (−0.319 + 0.232i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.720 - 0.693i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.720 - 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(88\)    =    \(2^{3} \cdot 11\)
Sign: $-0.720 - 0.693i$
Analytic conductor: \(14.1137\)
Root analytic conductor: \(3.75683\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{88} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 88,\ (\ :5/2),\ -0.720 - 0.693i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.882688 + 2.19193i\)
\(L(\frac12)\) \(\approx\) \(0.882688 + 2.19193i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (79.9 + 393. i)T \)
good3 \( 1 + (-6.70 - 20.6i)T + (-196. + 142. i)T^{2} \)
5 \( 1 + (-67.3 - 48.9i)T + (965. + 2.97e3i)T^{2} \)
7 \( 1 + (74.4 - 229. i)T + (-1.35e4 - 9.87e3i)T^{2} \)
13 \( 1 + (-651. + 473. i)T + (1.14e5 - 3.53e5i)T^{2} \)
17 \( 1 + (-1.02e3 - 745. i)T + (4.38e5 + 1.35e6i)T^{2} \)
19 \( 1 + (564. + 1.73e3i)T + (-2.00e6 + 1.45e6i)T^{2} \)
23 \( 1 + 3.37e3T + 6.43e6T^{2} \)
29 \( 1 + (302. - 931. i)T + (-1.65e7 - 1.20e7i)T^{2} \)
31 \( 1 + (1.70e3 - 1.24e3i)T + (8.84e6 - 2.72e7i)T^{2} \)
37 \( 1 + (-371. + 1.14e3i)T + (-5.61e7 - 4.07e7i)T^{2} \)
41 \( 1 + (1.86e3 + 5.74e3i)T + (-9.37e7 + 6.80e7i)T^{2} \)
43 \( 1 - 1.97e4T + 1.47e8T^{2} \)
47 \( 1 + (-5.93e3 - 1.82e4i)T + (-1.85e8 + 1.34e8i)T^{2} \)
53 \( 1 + (-1.36e4 + 9.89e3i)T + (1.29e8 - 3.97e8i)T^{2} \)
59 \( 1 + (-1.64e3 + 5.07e3i)T + (-5.78e8 - 4.20e8i)T^{2} \)
61 \( 1 + (-7.13e3 - 5.18e3i)T + (2.60e8 + 8.03e8i)T^{2} \)
67 \( 1 + 2.59e4T + 1.35e9T^{2} \)
71 \( 1 + (-2.57e4 - 1.87e4i)T + (5.57e8 + 1.71e9i)T^{2} \)
73 \( 1 + (-1.36e3 + 4.20e3i)T + (-1.67e9 - 1.21e9i)T^{2} \)
79 \( 1 + (-5.23e4 + 3.80e4i)T + (9.50e8 - 2.92e9i)T^{2} \)
83 \( 1 + (2.83e4 + 2.05e4i)T + (1.21e9 + 3.74e9i)T^{2} \)
89 \( 1 + 5.00e4T + 5.58e9T^{2} \)
97 \( 1 + (2.72e4 - 1.98e4i)T + (2.65e9 - 8.16e9i)T^{2} \)
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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

−13.84764416689358537674727777396, −12.65326923199955740394956527666, −11.02690930373063130412299282091, −10.22865745785398871442772506602, −9.274123664750092932637412448684, −8.506488033690131395685645540095, −6.08019444520415517539609438874, −5.61350910048538822985528472716, −3.42849610833683171284323889566, −2.52066265795792516445212312656, 0.978050462461175040718705402205, 1.89554073192896241350071476252, 4.08206173985093145179157034651, 5.99176857028319611420413871735, 7.05525062866968332499176621323, 8.036141011034969045608498542197, 9.514669932988743253116735438471, 10.33657038470424756916539240453, 12.20529137838960248473174727625, 13.02840311525799671432275746978

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