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Properties

Label 2-578-17.13-c1-0-7
Degree $2$
Conductor $578$
Sign $0.959 + 0.281i$
Analytic cond. $4.61535$
Root an. cond. $2.14833$
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  − i·2-s + (0.376 + 0.376i)3-s − 4-s + (2.74 + 2.74i)5-s + (0.376 − 0.376i)6-s + (1.54 − 1.54i)7-s + i·8-s − 2.71i·9-s + (2.74 − 2.74i)10-s + (−1.16 + 1.16i)11-s + (−0.376 − 0.376i)12-s + 0.588·13-s + (−1.54 − 1.54i)14-s + 2.06i·15-s + 16-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.217 + 0.217i)3-s − 0.5·4-s + (1.22 + 1.22i)5-s + (0.153 − 0.153i)6-s + (0.583 − 0.583i)7-s + 0.353i·8-s − 0.905i·9-s + (0.867 − 0.867i)10-s + (−0.352 + 0.352i)11-s + (−0.108 − 0.108i)12-s + 0.163·13-s + (−0.412 − 0.412i)14-s + 0.532i·15-s + 0.250·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.281i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 + 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(578\)    =    \(2 \cdot 17^{2}\)
Sign: $0.959 + 0.281i$
Analytic conductor: \(4.61535\)
Root analytic conductor: \(2.14833\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{578} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 578,\ (\ :1/2),\ 0.959 + 0.281i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90398 - 0.273615i\)
\(L(\frac12)\) \(\approx\) \(1.90398 - 0.273615i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
17 \( 1 \)
good3 \( 1 + (-0.376 - 0.376i)T + 3iT^{2} \)
5 \( 1 + (-2.74 - 2.74i)T + 5iT^{2} \)
7 \( 1 + (-1.54 + 1.54i)T - 7iT^{2} \)
11 \( 1 + (1.16 - 1.16i)T - 11iT^{2} \)
13 \( 1 - 0.588T + 13T^{2} \)
19 \( 1 - 6.80iT - 19T^{2} \)
23 \( 1 + (-5.77 + 5.77i)T - 23iT^{2} \)
29 \( 1 + (-4.10 - 4.10i)T + 29iT^{2} \)
31 \( 1 + (2.90 + 2.90i)T + 31iT^{2} \)
37 \( 1 + (-1.48 - 1.48i)T + 37iT^{2} \)
41 \( 1 + (1.86 - 1.86i)T - 41iT^{2} \)
43 \( 1 + 4.87iT - 43T^{2} \)
47 \( 1 + 8.47T + 47T^{2} \)
53 \( 1 + 6.98iT - 53T^{2} \)
59 \( 1 + 7.84iT - 59T^{2} \)
61 \( 1 + (0.130 - 0.130i)T - 61iT^{2} \)
67 \( 1 + 6.94T + 67T^{2} \)
71 \( 1 + (0.0350 + 0.0350i)T + 71iT^{2} \)
73 \( 1 + (1.25 + 1.25i)T + 73iT^{2} \)
79 \( 1 + (0.261 - 0.261i)T - 79iT^{2} \)
83 \( 1 - 2.67iT - 83T^{2} \)
89 \( 1 - 4.16T + 89T^{2} \)
97 \( 1 + (-9.19 - 9.19i)T + 97iT^{2} \)
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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

−10.49983847626664403624615817769, −10.07841541585731876113477915882, −9.256869345955846939985921605950, −8.173837480243400108510753269850, −6.95401539815815711375483359398, −6.18119193860788347525971573399, −4.98335558373628883639598829283, −3.69135335369295181790351830953, −2.76650782427730192264222819110, −1.54209434160225361607872280971, 1.36861630167947697832345460455, 2.64511473329656760247763122654, 4.78873214959723536144827057281, 5.15582501748525810289672478380, 6.00834525528280184922269294174, 7.25451352171461102141356293015, 8.278345531296874956149115306496, 8.870885252505244942090448915413, 9.498266731753205731716848091314, 10.65126491064092795490235912810

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