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Properties

Label 2-1088-136.123-c0-0-1
Degree $2$
Conductor $1088$
Sign $0.992 + 0.122i$
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 − i)7-s + i·9-s + i·17-s + (1 − i)23-s − i·25-s + (−1 − i)31-s + (1 + i)41-s + 2i·47-s − i·49-s + (1 + i)63-s + (−1 − i)71-s + (1 − i)73-s + (−1 + i)79-s − 81-s − 2·89-s + ⋯
L(s)  = 1  + (1 − i)7-s + i·9-s + i·17-s + (1 − i)23-s − i·25-s + (−1 − i)31-s + (1 + i)41-s + 2i·47-s − i·49-s + (1 + i)63-s + (−1 − i)71-s + (1 − i)73-s + (−1 + i)79-s − 81-s − 2·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1088\)    =    \(2^{6} \cdot 17\)
Sign: $0.992 + 0.122i$
Analytic conductor: \(0.542982\)
Root analytic conductor: \(0.736873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1088} (735, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1088,\ (\ :0),\ 0.992 + 0.122i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.152492549\)
\(L(\frac12)\) \(\approx\) \(1.152492549\)
\(L(1)\) 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 \( 1 - iT^{2} \)
5 \( 1 + iT^{2} \)
7 \( 1 + (-1 + i)T - iT^{2} \)
11 \( 1 + iT^{2} \)
13 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-1 + i)T - iT^{2} \)
29 \( 1 + iT^{2} \)
31 \( 1 + (1 + i)T + iT^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + (-1 - i)T + iT^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - 2iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - iT^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (1 + i)T + iT^{2} \)
73 \( 1 + (-1 + i)T - iT^{2} \)
79 \( 1 + (1 - i)T - iT^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + 2T + T^{2} \)
97 \( 1 + (1 - i)T - iT^{2} \)
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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

−10.29539027160360289576905360868, −9.207104212063457342403818983172, −8.089880113597310968792268281539, −7.83319433846210301241579574043, −6.80156880159838407472065122574, −5.75769980492948676887986146208, −4.64488068463279058962367993754, −4.17934846345076690025045127966, −2.63091415267530276567261972783, −1.41063748802897300466531527802, 1.48006154154857299071007424700, 2.79499827300565079058612742474, 3.84451478153896636410207710267, 5.20686665576613758111267118810, 5.53679234791846772503004044636, 6.88433657484132285083816052882, 7.49619915812520977467354850866, 8.792312373527920006622557127814, 8.993251950163027164615968774469, 9.946646898343575098733340442893

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