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Properties

Label 1-517-517.516-r0-0-0
Degree $1$
Conductor $517$
Sign $1$
Analytic cond. $2.40093$
Root an. cond. $2.40093$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 5-s − 6-s − 7-s − 8-s + 9-s + 10-s + 12-s + 13-s + 14-s − 15-s + 16-s − 17-s − 18-s + 19-s − 20-s − 21-s − 23-s − 24-s + 25-s − 26-s + 27-s − 28-s + 29-s + 30-s + ⋯
L(s)  = 1  − 2-s + 3-s + 4-s − 5-s − 6-s − 7-s − 8-s + 9-s + 10-s + 12-s + 13-s + 14-s − 15-s + 16-s − 17-s − 18-s + 19-s − 20-s − 21-s − 23-s − 24-s + 25-s − 26-s + 27-s − 28-s + 29-s + 30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(517\)    =    \(11 \cdot 47\)
Sign: $1$
Analytic conductor: \(2.40093\)
Root analytic conductor: \(2.40093\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{517} (516, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 517,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9605553244\)
\(L(\frac12)\) \(\approx\) \(0.9605553244\)
\(L(1)\) \(\approx\) \(0.8150416046\)
\(L(1)\) \(\approx\) \(0.8150416046\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
47 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 - T \)
13 \( 1 + T \)
17 \( 1 - T \)
19 \( 1 + T \)
23 \( 1 - T \)
29 \( 1 + T \)
31 \( 1 - T \)
37 \( 1 + T \)
41 \( 1 + T \)
43 \( 1 + T \)
53 \( 1 + T \)
59 \( 1 + T \)
61 \( 1 - T \)
67 \( 1 - T \)
71 \( 1 + T \)
73 \( 1 + T \)
79 \( 1 - T \)
83 \( 1 - T \)
89 \( 1 + T \)
97 \( 1 + T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.87871406316782185673218072247, −22.7591217759394260909887084613, −21.66949836669558016578251778127, −20.5703943963632118211257803545, −19.83638560295703373205421205127, −19.59789992867365355892071174250, −18.51910973492523613380884168993, −18.04564590039128722500258894180, −16.41416165145175919936157634978, −15.89554012746781279682982220566, −15.43779767358639572836377617394, −14.274210838838828922718744481795, −13.15106739301593195486545099013, −12.266600529821474785128169668694, −11.2311778776625158090077458275, −10.29480683019232120469411987164, −9.305699726217742196441757796824, −8.68822171090208642315119255085, −7.78558839193715476058884510655, −7.05074753687007601338979181334, −6.07907459752588685523953447651, −4.134720771013549741010607730717, −3.33251005245750670482742925856, −2.40444214031089452904432127466, −0.90961621870056013122782460926, 0.90961621870056013122782460926, 2.40444214031089452904432127466, 3.33251005245750670482742925856, 4.134720771013549741010607730717, 6.07907459752588685523953447651, 7.05074753687007601338979181334, 7.78558839193715476058884510655, 8.68822171090208642315119255085, 9.305699726217742196441757796824, 10.29480683019232120469411987164, 11.2311778776625158090077458275, 12.266600529821474785128169668694, 13.15106739301593195486545099013, 14.274210838838828922718744481795, 15.43779767358639572836377617394, 15.89554012746781279682982220566, 16.41416165145175919936157634978, 18.04564590039128722500258894180, 18.51910973492523613380884168993, 19.59789992867365355892071174250, 19.83638560295703373205421205127, 20.5703943963632118211257803545, 21.66949836669558016578251778127, 22.7591217759394260909887084613, 23.87871406316782185673218072247

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