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Properties

Label 2-1088-136.45-c0-0-1
Degree $2$
Conductor $1088$
Sign $0.968 + 0.250i$
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  + (1.08 − 0.216i)3-s + (0.216 − 0.0897i)9-s + (0.0761 − 0.382i)11-s + (0.923 + 0.382i)17-s + (0.541 − 1.30i)19-s + (0.382 + 0.923i)25-s + (−0.707 + 0.472i)27-s − 0.433i·33-s + (−1.63 + 1.08i)41-s + (−0.707 − 1.70i)43-s + (−0.382 + 0.923i)49-s + (1.08 + 0.216i)51-s + (0.306 − 1.54i)57-s + (−1.70 + 0.707i)59-s − 0.765·67-s + ⋯
L(s)  = 1  + (1.08 − 0.216i)3-s + (0.216 − 0.0897i)9-s + (0.0761 − 0.382i)11-s + (0.923 + 0.382i)17-s + (0.541 − 1.30i)19-s + (0.382 + 0.923i)25-s + (−0.707 + 0.472i)27-s − 0.433i·33-s + (−1.63 + 1.08i)41-s + (−0.707 − 1.70i)43-s + (−0.382 + 0.923i)49-s + (1.08 + 0.216i)51-s + (0.306 − 1.54i)57-s + (−1.70 + 0.707i)59-s − 0.765·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1088\)    =    \(2^{6} \cdot 17\)
Sign: $0.968 + 0.250i$
Analytic conductor: \(0.542982\)
Root analytic conductor: \(0.736873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1088} (929, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1088,\ (\ :0),\ 0.968 + 0.250i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.449783289\)
\(L(\frac12)\) \(\approx\) \(1.449783289\)
\(L(1)\) 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 \( 1 + (-0.923 - 0.382i)T \)
good3 \( 1 + (-1.08 + 0.216i)T + (0.923 - 0.382i)T^{2} \)
5 \( 1 + (-0.382 - 0.923i)T^{2} \)
7 \( 1 + (0.382 - 0.923i)T^{2} \)
11 \( 1 + (-0.0761 + 0.382i)T + (-0.923 - 0.382i)T^{2} \)
13 \( 1 + iT^{2} \)
19 \( 1 + (-0.541 + 1.30i)T + (-0.707 - 0.707i)T^{2} \)
23 \( 1 + (-0.923 - 0.382i)T^{2} \)
29 \( 1 + (0.382 + 0.923i)T^{2} \)
31 \( 1 + (0.923 - 0.382i)T^{2} \)
37 \( 1 + (0.923 - 0.382i)T^{2} \)
41 \( 1 + (1.63 - 1.08i)T + (0.382 - 0.923i)T^{2} \)
43 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (-0.707 - 0.707i)T^{2} \)
59 \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \)
61 \( 1 + (0.382 - 0.923i)T^{2} \)
67 \( 1 + 0.765T + T^{2} \)
71 \( 1 + (-0.923 + 0.382i)T^{2} \)
73 \( 1 + (-0.324 - 0.216i)T + (0.382 + 0.923i)T^{2} \)
79 \( 1 + (0.923 + 0.382i)T^{2} \)
83 \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \)
89 \( 1 + (1.30 + 1.30i)T + iT^{2} \)
97 \( 1 + (-0.923 + 1.38i)T + (-0.382 - 0.923i)T^{2} \)
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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

−9.830691431450316355612156584509, −9.063567433506825619729417210608, −8.468452932693587088859159719723, −7.64463687062846948210733663957, −6.93305754948833650905377741744, −5.76474820119756197298069245908, −4.82832755549310063127547586626, −3.46048502433788796522327325968, −2.91420156371920311355969062279, −1.56163472115947424439604959987, 1.72580695048995160415166461382, 2.98779624154534973455047546949, 3.66604782685401967110424476136, 4.80714876299460668954882558605, 5.84193264748536161077541417510, 6.89486458245479011355292523572, 7.955145347325006046061441311512, 8.290799336928387505990021133479, 9.399354489243330920940190056403, 9.848082245471008142406897645837

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