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Properties

Label 2-544-17.4-c1-0-2
Degree $2$
Conductor $544$
Sign $-0.237 - 0.971i$
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  + (−1.87 + 1.87i)3-s + (1 − i)5-s + (−1.18 − 1.18i)7-s − 4.06i·9-s + (3.87 + 3.87i)11-s + 3.06·13-s + 3.75i·15-s + (−1.69 + 3.75i)17-s + 3.75i·19-s + 4.45·21-s + (−3.18 − 3.18i)23-s + 3i·25-s + (2.00 + 2.00i)27-s + (−4.06 + 4.06i)29-s + (−2.94 + 2.94i)31-s + ⋯
L(s)  = 1  + (−1.08 + 1.08i)3-s + (0.447 − 0.447i)5-s + (−0.447 − 0.447i)7-s − 1.35i·9-s + (1.16 + 1.16i)11-s + 0.849·13-s + 0.970i·15-s + (−0.410 + 0.911i)17-s + 0.862i·19-s + 0.971·21-s + (−0.664 − 0.664i)23-s + 0.600i·25-s + (0.384 + 0.384i)27-s + (−0.754 + 0.754i)29-s + (−0.528 + 0.528i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 544 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.237 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 544 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.237 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(544\)    =    \(2^{5} \cdot 17\)
Sign: $-0.237 - 0.971i$
Analytic conductor: \(4.34386\)
Root analytic conductor: \(2.08419\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{544} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 544,\ (\ :1/2),\ -0.237 - 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.590899 + 0.752442i\)
\(L(\frac12)\) \(\approx\) \(0.590899 + 0.752442i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.69 - 3.75i)T \)
good3 \( 1 + (1.87 - 1.87i)T - 3iT^{2} \)
5 \( 1 + (-1 + i)T - 5iT^{2} \)
7 \( 1 + (1.18 + 1.18i)T + 7iT^{2} \)
11 \( 1 + (-3.87 - 3.87i)T + 11iT^{2} \)
13 \( 1 - 3.06T + 13T^{2} \)
19 \( 1 - 3.75iT - 19T^{2} \)
23 \( 1 + (3.18 + 3.18i)T + 23iT^{2} \)
29 \( 1 + (4.06 - 4.06i)T - 29iT^{2} \)
31 \( 1 + (2.94 - 2.94i)T - 31iT^{2} \)
37 \( 1 + (-2.38 + 2.38i)T - 37iT^{2} \)
41 \( 1 + (0.389 + 0.389i)T + 41iT^{2} \)
43 \( 1 - 1.63iT - 43T^{2} \)
47 \( 1 - 10.1T + 47T^{2} \)
53 \( 1 - 13.6iT - 53T^{2} \)
59 \( 1 - 9.88iT - 59T^{2} \)
61 \( 1 + (-7.12 - 7.12i)T + 61iT^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + (-2.57 + 2.57i)T - 71iT^{2} \)
73 \( 1 + (6.51 - 6.51i)T - 73iT^{2} \)
79 \( 1 + (-3.42 - 3.42i)T + 79iT^{2} \)
83 \( 1 + 13.1iT - 83T^{2} \)
89 \( 1 - 15.9T + 89T^{2} \)
97 \( 1 + (1.45 - 1.45i)T - 97iT^{2} \)
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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

−10.77507267722964254499691078760, −10.33585872850727904002955081955, −9.445367725025174432557889415232, −8.812712832977373763927338246061, −7.22806574553595352443745604306, −6.20177202084446437077734827382, −5.57544353037549708631181884877, −4.27807217771233085149224763317, −3.87265047939511365458164988100, −1.52221503240498150236664846967, 0.68158061468611736175352371980, 2.20118418843896116116426050559, 3.67664906002982264455911023033, 5.36380907625798306458656334704, 6.33078730474736549834963922153, 6.42399018187125665672605135465, 7.62358988757402048158374792634, 8.838888442756973429213358730756, 9.632692264615915036114962307806, 10.99418642263886006501680355197

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