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Properties

Label 2-88-88.59-c2-0-14
Degree $2$
Conductor $88$
Sign $0.973 + 0.226i$
Analytic cond. $2.39782$
Root an. cond. $1.54849$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.61 + 1.17i)2-s + (1.52 − 4.70i)3-s + (1.23 + 3.80i)4-s + (7.99 − 5.81i)6-s + (−2.47 + 7.60i)8-s + (−12.4 − 9.07i)9-s + (10.2 − 4.03i)11-s + 19.7·12-s + (−12.9 + 9.40i)16-s + (−26.6 + 19.3i)17-s + (−9.53 − 29.3i)18-s + (−8.23 + 25.3i)19-s + (21.3 + 5.50i)22-s + (31.9 + 23.2i)24-s + (7.72 − 23.7i)25-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (0.509 − 1.56i)3-s + (0.309 + 0.951i)4-s + (1.33 − 0.968i)6-s + (−0.309 + 0.951i)8-s + (−1.38 − 1.00i)9-s + (0.930 − 0.366i)11-s + 1.64·12-s + (−0.809 + 0.587i)16-s + (−1.56 + 1.13i)17-s + (−0.529 − 1.63i)18-s + (−0.433 + 1.33i)19-s + (0.968 + 0.250i)22-s + (1.33 + 0.968i)24-s + (0.309 − 0.951i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.226i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.973 + 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(88\)    =    \(2^{3} \cdot 11\)
Sign: $0.973 + 0.226i$
Analytic conductor: \(2.39782\)
Root analytic conductor: \(1.54849\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{88} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 88,\ (\ :1),\ 0.973 + 0.226i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.15773 - 0.248067i\)
\(L(\frac12)\) \(\approx\) \(2.15773 - 0.248067i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.61 - 1.17i)T \)
11 \( 1 + (-10.2 + 4.03i)T \)
good3 \( 1 + (-1.52 + 4.70i)T + (-7.28 - 5.29i)T^{2} \)
5 \( 1 + (-7.72 + 23.7i)T^{2} \)
7 \( 1 + (39.6 - 28.8i)T^{2} \)
13 \( 1 + (-52.2 - 160. i)T^{2} \)
17 \( 1 + (26.6 - 19.3i)T + (89.3 - 274. i)T^{2} \)
19 \( 1 + (8.23 - 25.3i)T + (-292. - 212. i)T^{2} \)
23 \( 1 - 529T^{2} \)
29 \( 1 + (680. - 494. i)T^{2} \)
31 \( 1 + (-296. - 913. i)T^{2} \)
37 \( 1 + (1.10e3 - 804. i)T^{2} \)
41 \( 1 + (-23.8 + 73.3i)T + (-1.35e3 - 988. i)T^{2} \)
43 \( 1 - 38.5T + 1.84e3T^{2} \)
47 \( 1 + (1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (-868. - 2.67e3i)T^{2} \)
59 \( 1 + (32.7 + 100. i)T + (-2.81e3 + 2.04e3i)T^{2} \)
61 \( 1 + (-1.14e3 + 3.53e3i)T^{2} \)
67 \( 1 - 132.T + 4.48e3T^{2} \)
71 \( 1 + (-1.55e3 + 4.79e3i)T^{2} \)
73 \( 1 + (-29.3 - 90.2i)T + (-4.31e3 + 3.13e3i)T^{2} \)
79 \( 1 + (-1.92e3 - 5.93e3i)T^{2} \)
83 \( 1 + (-79.2 + 57.5i)T + (2.12e3 - 6.55e3i)T^{2} \)
89 \( 1 + 177.T + 7.92e3T^{2} \)
97 \( 1 + (-19.1 - 13.9i)T + (2.90e3 + 8.94e3i)T^{2} \)
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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

−14.01148218711692903061276323475, −12.80458683110274919550845521427, −12.37893761004400349175681965464, −11.10804833077334027858421809628, −8.772612592841555937417722514792, −8.030889874284857452421473816126, −6.74268073573995078955348110744, −6.10335970317799283578348180406, −3.92547442557286396912851418650, −2.10453548369215504504679945370, 2.74992390597921100078301095032, 4.18632406992562719943464980388, 4.94604373171545802322980492348, 6.73892300498682766426994759019, 9.065848489256704239230738093241, 9.535598826998043303944554415003, 10.89547200858715446909229106510, 11.48178064265543272854261514410, 13.08970928093494874380144199931, 14.07509652763341228315297394994

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