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Properties

Label 2-513-171.106-c1-0-15
Degree $2$
Conductor $513$
Sign $-0.978 + 0.208i$
Analytic cond. $4.09632$
Root an. cond. $2.02393$
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  + (−0.847 − 1.46i)2-s + (−0.435 + 0.753i)4-s − 0.0882·5-s + (1.84 − 3.19i)7-s − 1.91·8-s + (0.0747 + 0.129i)10-s + (1.97 − 3.42i)11-s + (−2.03 + 3.51i)13-s − 6.25·14-s + (2.49 + 4.31i)16-s + (0.586 − 1.01i)17-s + (3.26 − 2.89i)19-s + (0.0383 − 0.0664i)20-s − 6.69·22-s + (1.91 − 3.31i)23-s + ⋯
L(s)  = 1  + (−0.599 − 1.03i)2-s + (−0.217 + 0.376i)4-s − 0.0394·5-s + (0.698 − 1.20i)7-s − 0.676·8-s + (0.0236 + 0.0409i)10-s + (0.596 − 1.03i)11-s + (−0.563 + 0.975i)13-s − 1.67·14-s + (0.622 + 1.07i)16-s + (0.142 − 0.246i)17-s + (0.748 − 0.663i)19-s + (0.00858 − 0.0148i)20-s − 1.42·22-s + (0.399 − 0.691i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.208i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 + 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(513\)    =    \(3^{3} \cdot 19\)
Sign: $-0.978 + 0.208i$
Analytic conductor: \(4.09632\)
Root analytic conductor: \(2.02393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{513} (505, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 513,\ (\ :1/2),\ -0.978 + 0.208i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0999668 - 0.950552i\)
\(L(\frac12)\) \(\approx\) \(0.0999668 - 0.950552i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 + (-3.26 + 2.89i)T \)
good2 \( 1 + (0.847 + 1.46i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 0.0882T + 5T^{2} \)
7 \( 1 + (-1.84 + 3.19i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.97 + 3.42i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.03 - 3.51i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.586 + 1.01i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.91 + 3.31i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.56T + 29T^{2} \)
31 \( 1 + (4.14 + 7.18i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.88T + 37T^{2} \)
41 \( 1 - 4.66T + 41T^{2} \)
43 \( 1 + (-4.12 - 7.14i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 4.43T + 47T^{2} \)
53 \( 1 + (-3.62 - 6.27i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 1.01T + 59T^{2} \)
61 \( 1 - 3.22T + 61T^{2} \)
67 \( 1 + (-1.45 + 2.51i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.36 + 7.56i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.43 + 5.95i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.65 - 4.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.34 - 5.80i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.41 + 7.64i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.894 + 1.55i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.67702562290870706460047320129, −9.580961021014075797010729931751, −9.102559028757520176606412185818, −7.902519845960808640105556300833, −7.01867551320666795510818678099, −5.81499662587912027112774329473, −4.40357296612951899416509040972, −3.42383374243175122233057957457, −1.96173482235622633249128033256, −0.69730142603218416045478506218, 1.96318662304198389077702228166, 3.49912460407169198157402975717, 5.28125246684074335390556115488, 5.66385802850130560939014790975, 7.05763948622977666028253572917, 7.65220753290682694059830084106, 8.523893719704852784230657052139, 9.301712416130491931504393885835, 10.04466050797062736475018277503, 11.43712127892506136540391318089

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