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Properties

Label 2-544-32.5-c1-0-5
Degree $2$
Conductor $544$
Sign $-0.993 + 0.118i$
Analytic cond. $4.34386$
Root an. cond. $2.08419$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.24 − 0.661i)2-s + (1.13 + 2.75i)3-s + (1.12 + 1.65i)4-s + (−3.02 − 1.25i)5-s + (0.395 − 4.19i)6-s + (1.22 + 1.22i)7-s + (−0.311 − 2.81i)8-s + (−4.15 + 4.15i)9-s + (2.94 + 3.56i)10-s + (1.01 − 2.45i)11-s + (−3.26 + 4.97i)12-s + (−5.68 + 2.35i)13-s + (−0.720 − 2.34i)14-s − 9.74i·15-s + (−1.47 + 3.71i)16-s + i·17-s + ⋯
L(s)  = 1  + (−0.883 − 0.467i)2-s + (0.658 + 1.58i)3-s + (0.562 + 0.826i)4-s + (−1.35 − 0.559i)5-s + (0.161 − 1.71i)6-s + (0.462 + 0.462i)7-s + (−0.110 − 0.993i)8-s + (−1.38 + 1.38i)9-s + (0.932 + 1.12i)10-s + (0.306 − 0.740i)11-s + (−0.943 + 1.43i)12-s + (−1.57 + 0.653i)13-s + (−0.192 − 0.625i)14-s − 2.51i·15-s + (−0.367 + 0.929i)16-s + 0.242i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 544 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.118i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 544 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(544\)    =    \(2^{5} \cdot 17\)
Sign: $-0.993 + 0.118i$
Analytic conductor: \(4.34386\)
Root analytic conductor: \(2.08419\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{544} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 544,\ (\ :1/2),\ -0.993 + 0.118i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0202279 - 0.341518i\)
\(L(\frac12)\) \(\approx\) \(0.0202279 - 0.341518i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.24 + 0.661i)T \)
17 \( 1 - iT \)
good3 \( 1 + (-1.13 - 2.75i)T + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (3.02 + 1.25i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (-1.22 - 1.22i)T + 7iT^{2} \)
11 \( 1 + (-1.01 + 2.45i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (5.68 - 2.35i)T + (9.19 - 9.19i)T^{2} \)
19 \( 1 + (0.386 - 0.160i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (5.00 - 5.00i)T - 23iT^{2} \)
29 \( 1 + (2.60 + 6.27i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + 1.80T + 31T^{2} \)
37 \( 1 + (2.25 + 0.933i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (1.58 - 1.58i)T - 41iT^{2} \)
43 \( 1 + (-1.74 + 4.22i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + 9.11iT - 47T^{2} \)
53 \( 1 + (2.36 - 5.70i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (0.0181 + 0.00751i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (1.33 + 3.22i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (-4.24 - 10.2i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (-11.7 - 11.7i)T + 71iT^{2} \)
73 \( 1 + (10.2 - 10.2i)T - 73iT^{2} \)
79 \( 1 - 13.3iT - 79T^{2} \)
83 \( 1 + (8.52 - 3.53i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-4.21 - 4.21i)T + 89iT^{2} \)
97 \( 1 - 1.25T + 97T^{2} \)
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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

−11.38711701557688167437490197831, −10.10650270382132165712260846250, −9.535683880901083359167087174454, −8.646415050473133913502436803160, −8.213504198980729262382656796386, −7.30207562153652490500298446964, −5.36487053546038668056354658101, −4.16807789068388809935026957922, −3.71393897925154255282042215635, −2.34385843776444191714716060738, 0.23008373328614007334098614074, 1.85904721990508385285417219758, 3.01082968803341806865030492490, 4.72317008545098199808359800018, 6.39112936265995933148602207744, 7.24349448093347441633562786699, 7.59133865819940947707906169202, 8.085919113115476545850004227985, 9.157052515883107475684407425871, 10.31920925239101985922227191160

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