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Properties

Label 2-1088-136.133-c0-0-0
Degree $2$
Conductor $1088$
Sign $0.250 + 0.968i$
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

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

Invariants

Degree: \(2\)
Conductor: \(1088\)    =    \(2^{6} \cdot 17\)
Sign: $0.250 + 0.968i$
Analytic conductor: \(0.542982\)
Root analytic conductor: \(0.736873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1088} (609, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1088,\ (\ :0),\ 0.250 + 0.968i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6257771433\)
\(L(\frac12)\) \(\approx\) \(0.6257771433\)
\(L(1)\) 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

−10.17038181227685752845492432389, −8.999804835440467185311141574192, −8.326630118057965322236225546694, −7.09441272631478323409299105652, −6.58357922829982331804084264812, −5.53042233607122167010412921404, −5.02385233660886735846766305987, −3.73202576326616159927047456584, −2.46025773245760664283383299983, −0.69852590482989027275259853024, 1.50432967335632638434392299463, 3.13050754418479503238696320847, 4.29264292068983120085730101996, 5.22420779286927274313362658930, 5.91552499042324007308267869262, 6.69322256853260984163006164851, 7.78238459245791514324633212125, 8.501854685644893012426073277972, 9.776749940916230511143994597195, 10.19397000757716866614212797492

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