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Properties

Label 2-1088-136.125-c0-0-0
Degree $2$
Conductor $1088$
Sign $-0.204 - 0.978i$
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 + 1.63i)3-s + (−1.63 + 0.675i)9-s + (1.92 + 0.382i)11-s + (−0.923 − 0.382i)17-s + (−0.541 + 1.30i)19-s + (−0.382 − 0.923i)25-s + (−0.707 − 1.05i)27-s + 3.26i·33-s + (0.216 + 0.324i)41-s + (−0.707 − 1.70i)43-s + (0.382 − 0.923i)49-s + (0.324 − 1.63i)51-s + (−2.30 − 0.458i)57-s + (−1.70 + 0.707i)59-s + 0.765·67-s + ⋯
L(s)  = 1  + (0.324 + 1.63i)3-s + (−1.63 + 0.675i)9-s + (1.92 + 0.382i)11-s + (−0.923 − 0.382i)17-s + (−0.541 + 1.30i)19-s + (−0.382 − 0.923i)25-s + (−0.707 − 1.05i)27-s + 3.26i·33-s + (0.216 + 0.324i)41-s + (−0.707 − 1.70i)43-s + (0.382 − 0.923i)49-s + (0.324 − 1.63i)51-s + (−2.30 − 0.458i)57-s + (−1.70 + 0.707i)59-s + 0.765·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.204 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.204 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1088\)    =    \(2^{6} \cdot 17\)
Sign: $-0.204 - 0.978i$
Analytic conductor: \(0.542982\)
Root analytic conductor: \(0.736873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1088} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1088,\ (\ :0),\ -0.204 - 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.173147132\)
\(L(\frac12)\) \(\approx\) \(1.173147132\)
\(L(1)\) 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 \( 1 + (-0.324 - 1.63i)T + (-0.923 + 0.382i)T^{2} \)
5 \( 1 + (0.382 + 0.923i)T^{2} \)
7 \( 1 + (-0.382 + 0.923i)T^{2} \)
11 \( 1 + (-1.92 - 0.382i)T + (0.923 + 0.382i)T^{2} \)
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 \( 1 + (-0.216 - 0.324i)T + (-0.382 + 0.923i)T^{2} \)
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 \( 1 + (-1.08 + 1.63i)T + (-0.382 - 0.923i)T^{2} \)
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 \( 1 + (0.923 + 0.617i)T + (0.382 + 0.923i)T^{2} \)
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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

−10.19901062578974031207765882897, −9.441584552440296966513504679363, −8.947655961596254733375079719556, −8.165579699356822145522386675483, −6.81996355089682467493508656336, −6.01423869519503728391485251935, −4.82401072366003051773939726583, −4.08467860956650495343330929083, −3.54933951838268856491929288069, −2.04352750444639202145941728991, 1.17402827776802813127080373166, 2.16281856214574428888954764660, 3.37427629176284116471336989063, 4.54125142504142117715840547795, 6.07393684233369239700547571941, 6.56516291930919857878906405658, 7.19007260383785694595394255419, 8.161877494327783058340799197353, 8.928104871991970811634606990536, 9.430674926503093629457355416268

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