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Properties

Label 2-544-136.115-c0-0-0
Degree $2$
Conductor $544$
Sign $0.615 - 0.788i$
Analytic cond. $0.271491$
Root an. cond. $0.521048$
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)3-s + i·9-s + (−1 + i)11-s − i·17-s − 2i·19-s + i·25-s − 2·33-s + (−1 + i)41-s − i·49-s + (1 − i)51-s + (2 − 2i)57-s + (−1 − i)73-s + (−1 + i)75-s + 81-s + (1 + i)97-s + ⋯
L(s)  = 1  + (1 + i)3-s + i·9-s + (−1 + i)11-s − i·17-s − 2i·19-s + i·25-s − 2·33-s + (−1 + i)41-s − i·49-s + (1 − i)51-s + (2 − 2i)57-s + (−1 − i)73-s + (−1 + i)75-s + 81-s + (1 + i)97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 544 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 544 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(544\)    =    \(2^{5} \cdot 17\)
Sign: $0.615 - 0.788i$
Analytic conductor: \(0.271491\)
Root analytic conductor: \(0.521048\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{544} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 544,\ (\ :0),\ 0.615 - 0.788i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.149390037\)
\(L(\frac12)\) \(\approx\) \(1.149390037\)
\(L(1)\) 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 \( 1 + (-1 - i)T + iT^{2} \)
5 \( 1 - iT^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + (1 - i)T - iT^{2} \)
13 \( 1 - T^{2} \)
19 \( 1 + 2iT - T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (1 - i)T - iT^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + iT^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 + (1 + i)T + iT^{2} \)
79 \( 1 + iT^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T^{2} \)
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

−10.94650312861703973259013491336, −10.04383651830881850069606312777, −9.417967593017534616495470401710, −8.732910357168274800589377731277, −7.66387448801230824380541921874, −6.86327677469555973001738428462, −5.11553866225970619208072383982, −4.61711360990188590701757046471, −3.25570994334460187335919121106, −2.41362729761681242929427359539, 1.66694133518085799270266316944, 2.84280794624797988907724754719, 3.87688408736002465508446217170, 5.55461779392493099793135853753, 6.39602350338575691600831078639, 7.61358982587399219560266155137, 8.193259275441494264757268302151, 8.713421077775596253012788664652, 10.09094488240565540321952191972, 10.73996452382815104817666460372

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