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Properties

Label 2-88-8.5-c1-0-9
Degree $2$
Conductor $88$
Sign $-0.0788 + 0.996i$
Analytic cond. $0.702683$
Root an. cond. $0.838262$
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  + (0.739 − 1.20i)2-s − 2.35i·3-s + (−0.907 − 1.78i)4-s + 4.16i·5-s + (−2.83 − 1.73i)6-s + 0.933·7-s + (−2.81 − 0.223i)8-s − 2.53·9-s + (5.02 + 3.08i)10-s + i·11-s + (−4.19 + 2.13i)12-s − 2.93i·13-s + (0.689 − 1.12i)14-s + 9.80·15-s + (−2.35 + 3.23i)16-s + 2.44·17-s + ⋯
L(s)  = 1  + (0.522 − 0.852i)2-s − 1.35i·3-s + (−0.453 − 0.891i)4-s + 1.86i·5-s + (−1.15 − 0.709i)6-s + 0.352·7-s + (−0.996 − 0.0788i)8-s − 0.845·9-s + (1.58 + 0.974i)10-s + 0.301i·11-s + (−1.21 + 0.616i)12-s − 0.813i·13-s + (0.184 − 0.300i)14-s + 2.53·15-s + (−0.588 + 0.808i)16-s + 0.593·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0788 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0788 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(88\)    =    \(2^{3} \cdot 11\)
Sign: $-0.0788 + 0.996i$
Analytic conductor: \(0.702683\)
Root analytic conductor: \(0.838262\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{88} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 88,\ (\ :1/2),\ -0.0788 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.806334 - 0.872664i\)
\(L(\frac12)\) \(\approx\) \(0.806334 - 0.872664i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.739 + 1.20i)T \)
11 \( 1 - iT \)
good3 \( 1 + 2.35iT - 3T^{2} \)
5 \( 1 - 4.16iT - 5T^{2} \)
7 \( 1 - 0.933T + 7T^{2} \)
13 \( 1 + 2.93iT - 13T^{2} \)
17 \( 1 - 2.44T + 17T^{2} \)
19 \( 1 - 2.68iT - 19T^{2} \)
23 \( 1 + 3.47T + 23T^{2} \)
29 \( 1 - 4.57iT - 29T^{2} \)
31 \( 1 + 3.65T + 31T^{2} \)
37 \( 1 - 4.53iT - 37T^{2} \)
41 \( 1 - 4.12T + 41T^{2} \)
43 \( 1 + 11.4iT - 43T^{2} \)
47 \( 1 - 3.26T + 47T^{2} \)
53 \( 1 + 0.650iT - 53T^{2} \)
59 \( 1 + 2.90iT - 59T^{2} \)
61 \( 1 + 10.7iT - 61T^{2} \)
67 \( 1 - 5.42iT - 67T^{2} \)
71 \( 1 + 7.75T + 71T^{2} \)
73 \( 1 - 13.0T + 73T^{2} \)
79 \( 1 - 4.83T + 79T^{2} \)
83 \( 1 + 0.659iT - 83T^{2} \)
89 \( 1 + 2.74T + 89T^{2} \)
97 \( 1 - 1.82T + 97T^{2} \)
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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

−13.92296304037858766997452370929, −12.73076297875862983090971773164, −11.87547821949067548208917746951, −10.84277972882902542214597639872, −10.00771884630416997177276142272, −7.928039751354020858995347165509, −6.87144214785532843921905365448, −5.78040073698778140193136720194, −3.39884284659812330583731486452, −2.05548693400119437024034690665, 4.04959542342585581207760008239, 4.76062172302563492550400619817, 5.76311181345311865682848426844, 7.898385728354611120960560216940, 8.961094899937617811540220421716, 9.580684979772303504646201895289, 11.43936850086087383829905069451, 12.49139548745933837775526842980, 13.55605112429915540028156828690, 14.61825744528629099970781549093

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