Properties

Label 2-1088-68.19-c0-0-0
Degree $2$
Conductor $1088$
Sign $0.739 - 0.673i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 1.70i)5-s + (0.707 − 0.707i)9-s − 1.41i·13-s + i·17-s + (−1.70 + 1.70i)25-s + (0.292 + 0.707i)29-s + (−0.707 + 0.292i)37-s + (−0.292 + 0.707i)41-s + (1.70 + 0.707i)45-s + (−0.707 − 0.707i)49-s + (−1 − i)53-s + (0.707 − 1.70i)61-s + (2.41 − 1.00i)65-s + (0.292 + 0.707i)73-s − 1.00i·81-s + ⋯
L(s)  = 1  + (0.707 + 1.70i)5-s + (0.707 − 0.707i)9-s − 1.41i·13-s + i·17-s + (−1.70 + 1.70i)25-s + (0.292 + 0.707i)29-s + (−0.707 + 0.292i)37-s + (−0.292 + 0.707i)41-s + (1.70 + 0.707i)45-s + (−0.707 − 0.707i)49-s + (−1 − i)53-s + (0.707 − 1.70i)61-s + (2.41 − 1.00i)65-s + (0.292 + 0.707i)73-s − 1.00i·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1088\)    =    \(2^{6} \cdot 17\)
Sign: $0.739 - 0.673i$
Analytic conductor: \(0.542982\)
Root analytic conductor: \(0.736873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1088} (767, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1088,\ (\ :0),\ 0.739 - 0.673i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.204028640\)
\(L(\frac12)\) \(\approx\) \(1.204028640\)
\(L(1)\) 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 \( 1 - iT \)
good3 \( 1 + (-0.707 + 0.707i)T^{2} \)
5 \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \)
7 \( 1 + (0.707 + 0.707i)T^{2} \)
11 \( 1 + (-0.707 - 0.707i)T^{2} \)
13 \( 1 + 1.41iT - T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + (-0.707 - 0.707i)T^{2} \)
29 \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \)
31 \( 1 + (-0.707 + 0.707i)T^{2} \)
37 \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \)
41 \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (1 + i)T + iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.707 + 0.707i)T^{2} \)
73 \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \)
79 \( 1 + (-0.707 - 0.707i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + 1.41iT - T^{2} \)
97 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.14157946350446174712644323333, −9.735567167185059470290306289593, −8.453606198238965955869943384111, −7.53648746002321060518381097675, −6.66608437287159737844845453324, −6.20652038639220617237305203036, −5.16832660182240855445452260040, −3.62543208897488258804148033244, −3.06603280139541375563812241554, −1.76430190568777734531224439304, 1.34508858697204042613597973255, 2.27551395496187690721884307151, 4.16958569083548456535024817943, 4.75791733109002344681738599399, 5.50061292456226943874737950690, 6.58803543601726379848400787562, 7.56676357134748393937685000414, 8.488504426115647243413030115251, 9.276142860031979840965009840579, 9.658933927688974309156026261663

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