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Properties

Label 2-544-17.8-c1-0-10
Degree $2$
Conductor $544$
Sign $0.895 - 0.445i$
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  + (−0.433 + 1.04i)3-s + (2.16 + 0.898i)5-s + (3.20 − 1.32i)7-s + (1.21 + 1.21i)9-s + (−2.05 − 4.96i)11-s − 0.828i·13-s + (−1.87 + 1.87i)15-s + (3.79 + 1.60i)17-s + (1.01 − 1.01i)19-s + 3.92i·21-s + (−0.316 − 0.763i)23-s + (0.359 + 0.359i)25-s + (−4.93 + 2.04i)27-s + (−3.47 − 1.43i)29-s + (−3.20 + 7.74i)31-s + ⋯
L(s)  = 1  + (−0.250 + 0.603i)3-s + (0.969 + 0.401i)5-s + (1.21 − 0.502i)7-s + (0.405 + 0.405i)9-s + (−0.620 − 1.49i)11-s − 0.229i·13-s + (−0.485 + 0.485i)15-s + (0.921 + 0.389i)17-s + (0.232 − 0.232i)19-s + 0.857i·21-s + (−0.0659 − 0.159i)23-s + (0.0718 + 0.0718i)25-s + (−0.949 + 0.393i)27-s + (−0.644 − 0.267i)29-s + (−0.575 + 1.39i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 544 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.445i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 544 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 - 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(544\)    =    \(2^{5} \cdot 17\)
Sign: $0.895 - 0.445i$
Analytic conductor: \(4.34386\)
Root analytic conductor: \(2.08419\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{544} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 544,\ (\ :1/2),\ 0.895 - 0.445i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.74892 + 0.410840i\)
\(L(\frac12)\) \(\approx\) \(1.74892 + 0.410840i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 + (-3.79 - 1.60i)T \)
good3 \( 1 + (0.433 - 1.04i)T + (-2.12 - 2.12i)T^{2} \)
5 \( 1 + (-2.16 - 0.898i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (-3.20 + 1.32i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (2.05 + 4.96i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + 0.828iT - 13T^{2} \)
19 \( 1 + (-1.01 + 1.01i)T - 19iT^{2} \)
23 \( 1 + (0.316 + 0.763i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (3.47 + 1.43i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (3.20 - 7.74i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (1.37 - 3.32i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-10.1 + 4.18i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (-9.11 - 9.11i)T + 43iT^{2} \)
47 \( 1 + 3.75iT - 47T^{2} \)
53 \( 1 + (5.17 - 5.17i)T - 53iT^{2} \)
59 \( 1 + (5.86 + 5.86i)T + 59iT^{2} \)
61 \( 1 + (2.27 - 0.943i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + 8.93T + 67T^{2} \)
71 \( 1 + (5.29 - 12.7i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (1.86 + 0.773i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (3.56 + 8.60i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (-3.20 + 3.20i)T - 83iT^{2} \)
89 \( 1 - 0.313iT - 89T^{2} \)
97 \( 1 + (11.5 + 4.80i)T + (68.5 + 68.5i)T^{2} \)
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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

−10.76507318630177813990404163255, −10.28734848643693456513254776442, −9.252739902722554514633402248260, −8.120022619999253800452149245378, −7.44393747090689294973115869524, −5.94059458902036491971860989098, −5.39345020715728441085200813687, −4.32922280264581079582625717473, −2.98400430860754142169464081181, −1.46715782620397483139413059360, 1.48103695953062568765854755873, 2.23653251534753730571043111235, 4.27721754679442369717076322092, 5.32184037980524993745883141669, 5.94304651622133292774238894071, 7.40125257838857594868627237931, 7.72387709322936885293343003669, 9.262255572446306166145569565340, 9.627942841050126317542124526393, 10.76467390461862379785612327668

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