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Properties

Label 2-544-136.107-c1-0-8
Degree $2$
Conductor $544$
Sign $0.0820 + 0.996i$
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  + (−1.52 − 2.28i)3-s + (0.326 + 0.0649i)5-s + (3.74 − 0.745i)7-s + (−1.73 + 4.18i)9-s + (3.13 + 2.09i)11-s + (2.87 − 2.87i)13-s + (−0.349 − 0.843i)15-s + (4.11 + 0.311i)17-s + (−0.183 − 0.442i)19-s + (−7.41 − 7.41i)21-s + (−1.77 − 1.18i)23-s + (−4.51 − 1.87i)25-s + (4.12 − 0.821i)27-s + (−0.575 + 2.89i)29-s + (−4.13 − 6.18i)31-s + ⋯
L(s)  = 1  + (−0.880 − 1.31i)3-s + (0.145 + 0.0290i)5-s + (1.41 − 0.281i)7-s + (−0.578 + 1.39i)9-s + (0.943 + 0.630i)11-s + (0.797 − 0.797i)13-s + (−0.0902 − 0.217i)15-s + (0.997 + 0.0754i)17-s + (−0.0420 − 0.101i)19-s + (−1.61 − 1.61i)21-s + (−0.370 − 0.247i)23-s + (−0.903 − 0.374i)25-s + (0.794 − 0.158i)27-s + (−0.106 + 0.536i)29-s + (−0.741 − 1.11i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 544 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0820 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 544 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0820 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(544\)    =    \(2^{5} \cdot 17\)
Sign: $0.0820 + 0.996i$
Analytic conductor: \(4.34386\)
Root analytic conductor: \(2.08419\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{544} (175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 544,\ (\ :1/2),\ 0.0820 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00167 - 0.922622i\)
\(L(\frac12)\) \(\approx\) \(1.00167 - 0.922622i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.11 - 0.311i)T \)
good3 \( 1 + (1.52 + 2.28i)T + (-1.14 + 2.77i)T^{2} \)
5 \( 1 + (-0.326 - 0.0649i)T + (4.61 + 1.91i)T^{2} \)
7 \( 1 + (-3.74 + 0.745i)T + (6.46 - 2.67i)T^{2} \)
11 \( 1 + (-3.13 - 2.09i)T + (4.20 + 10.1i)T^{2} \)
13 \( 1 + (-2.87 + 2.87i)T - 13iT^{2} \)
19 \( 1 + (0.183 + 0.442i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (1.77 + 1.18i)T + (8.80 + 21.2i)T^{2} \)
29 \( 1 + (0.575 - 2.89i)T + (-26.7 - 11.0i)T^{2} \)
31 \( 1 + (4.13 + 6.18i)T + (-11.8 + 28.6i)T^{2} \)
37 \( 1 + (3.97 - 2.65i)T + (14.1 - 34.1i)T^{2} \)
41 \( 1 + (-1.10 - 5.57i)T + (-37.8 + 15.6i)T^{2} \)
43 \( 1 + (0.938 - 2.26i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + (-9.23 + 9.23i)T - 47iT^{2} \)
53 \( 1 + (-7.71 + 3.19i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (2.37 + 0.983i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (2.45 + 12.3i)T + (-56.3 + 23.3i)T^{2} \)
67 \( 1 - 13.0iT - 67T^{2} \)
71 \( 1 + (1.11 - 0.743i)T + (27.1 - 65.5i)T^{2} \)
73 \( 1 + (-0.580 + 2.91i)T + (-67.4 - 27.9i)T^{2} \)
79 \( 1 + (4.26 - 6.37i)T + (-30.2 - 72.9i)T^{2} \)
83 \( 1 + (3.77 - 1.56i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (9.04 - 9.04i)T - 89iT^{2} \)
97 \( 1 + (10.4 + 2.07i)T + (89.6 + 37.1i)T^{2} \)
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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

−10.90502962410751656287668473409, −9.905372961387198491428723891087, −8.462030239226294362957162513649, −7.77286027732724042416894499070, −7.02375381994648688923312786738, −5.99290183999270753506261198666, −5.28127683823659518944424067184, −3.96737559988481929959836044862, −1.94949528499354797821869962476, −1.07469655958675921683214856481, 1.52616429373705869334425501418, 3.67883958865642949063931018233, 4.37340226082515917878544143332, 5.51370669255253995233291836212, 5.94651630787165162488417642511, 7.44277947289200826813712769337, 8.704462229592021014399002470564, 9.232106600768730222320648911994, 10.34224978342947040280179044478, 11.03210262128670268460607678009

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