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Properties

Label 2-1088-17.11-c0-0-0
Degree $2$
Conductor $1088$
Sign $0.990 - 0.139i$
Analytic cond. $0.542982$
Root an. cond. $0.736873$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.324 + 0.216i)5-s + (0.923 − 0.382i)9-s + (−0.541 + 0.541i)13-s + (0.707 + 0.707i)17-s + (−0.324 − 0.783i)25-s + (0.923 − 1.38i)29-s + (0.216 + 1.08i)37-s + (−0.923 + 0.617i)41-s + (0.382 + 0.0761i)45-s + (−0.382 + 0.923i)49-s + (1.70 + 0.707i)53-s + (−1.08 − 1.63i)61-s + (−0.292 + 0.0582i)65-s + (−1.38 − 0.923i)73-s + (0.707 − 0.707i)81-s + ⋯
L(s)  = 1  + (0.324 + 0.216i)5-s + (0.923 − 0.382i)9-s + (−0.541 + 0.541i)13-s + (0.707 + 0.707i)17-s + (−0.324 − 0.783i)25-s + (0.923 − 1.38i)29-s + (0.216 + 1.08i)37-s + (−0.923 + 0.617i)41-s + (0.382 + 0.0761i)45-s + (−0.382 + 0.923i)49-s + (1.70 + 0.707i)53-s + (−1.08 − 1.63i)61-s + (−0.292 + 0.0582i)65-s + (−1.38 − 0.923i)73-s + (0.707 − 0.707i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1088\)    =    \(2^{6} \cdot 17\)
Sign: $0.990 - 0.139i$
Analytic conductor: \(0.542982\)
Root analytic conductor: \(0.736873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1088} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1088,\ (\ :0),\ 0.990 - 0.139i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.157244822\)
\(L(\frac12)\) \(\approx\) \(1.157244822\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (-0.923 + 0.382i)T^{2} \)
5 \( 1 + (-0.324 - 0.216i)T + (0.382 + 0.923i)T^{2} \)
7 \( 1 + (0.382 - 0.923i)T^{2} \)
11 \( 1 + (0.923 + 0.382i)T^{2} \)
13 \( 1 + (0.541 - 0.541i)T - iT^{2} \)
19 \( 1 + (0.707 + 0.707i)T^{2} \)
23 \( 1 + (-0.923 - 0.382i)T^{2} \)
29 \( 1 + (-0.923 + 1.38i)T + (-0.382 - 0.923i)T^{2} \)
31 \( 1 + (0.923 - 0.382i)T^{2} \)
37 \( 1 + (-0.216 - 1.08i)T + (-0.923 + 0.382i)T^{2} \)
41 \( 1 + (0.923 - 0.617i)T + (0.382 - 0.923i)T^{2} \)
43 \( 1 + (0.707 - 0.707i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \)
59 \( 1 + (-0.707 + 0.707i)T^{2} \)
61 \( 1 + (1.08 + 1.63i)T + (-0.382 + 0.923i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.923 + 0.382i)T^{2} \)
73 \( 1 + (1.38 + 0.923i)T + (0.382 + 0.923i)T^{2} \)
79 \( 1 + (0.923 + 0.382i)T^{2} \)
83 \( 1 + (-0.707 - 0.707i)T^{2} \)
89 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
97 \( 1 + (1.08 - 1.63i)T + (-0.382 - 0.923i)T^{2} \)
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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

−9.981010450941864953594319681163, −9.533059597892698967451668462525, −8.373012097692509071797947942411, −7.61518080621582378479883514665, −6.62528407115602782754379072250, −6.05768145287414962221395523799, −4.77132577506294604901225020387, −4.01051674633215824106562701620, −2.75309458499378129836174761988, −1.49903899223529298041775796364, 1.39482520063166848320630406195, 2.70283292829618732509908293865, 3.88895012150416553646375535002, 5.04288201662730100275397031587, 5.55538142576153194299366047549, 6.95199148896484360640282441689, 7.40370627099412628069144058634, 8.433234788332042852164135384385, 9.322495257517487560161210094492, 10.10865062615087516677173163491

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