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Properties

Label 2-538-269.268-c1-0-0
Degree $2$
Conductor $538$
Sign $0.409 + 0.912i$
Analytic cond. $4.29595$
Root an. cond. $2.07266$
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 + 2.84i·3-s − 4-s − 3.96·5-s − 2.84·6-s − 1.38i·7-s − i·8-s − 5.09·9-s − 3.96i·10-s − 0.0792·11-s − 2.84i·12-s + 2.95·13-s + 1.38·14-s − 11.2i·15-s + 16-s + 2.86i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.64i·3-s − 0.5·4-s − 1.77·5-s − 1.16·6-s − 0.524i·7-s − 0.353i·8-s − 1.69·9-s − 1.25i·10-s − 0.0238·11-s − 0.821i·12-s + 0.820·13-s + 0.370·14-s − 2.91i·15-s + 0.250·16-s + 0.694i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.409 + 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.409 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(538\)    =    \(2 \cdot 269\)
Sign: $0.409 + 0.912i$
Analytic conductor: \(4.29595\)
Root analytic conductor: \(2.07266\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{538} (537, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 538,\ (\ :1/2),\ 0.409 + 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0823430 - 0.0532831i\)
\(L(\frac12)\) \(\approx\) \(0.0823430 - 0.0532831i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
269 \( 1 + (14.9 - 6.71i)T \)
good3 \( 1 - 2.84iT - 3T^{2} \)
5 \( 1 + 3.96T + 5T^{2} \)
7 \( 1 + 1.38iT - 7T^{2} \)
11 \( 1 + 0.0792T + 11T^{2} \)
13 \( 1 - 2.95T + 13T^{2} \)
17 \( 1 - 2.86iT - 17T^{2} \)
19 \( 1 + 0.952iT - 19T^{2} \)
23 \( 1 + 8.78T + 23T^{2} \)
29 \( 1 + 9.47iT - 29T^{2} \)
31 \( 1 - 2.55iT - 31T^{2} \)
37 \( 1 + 5.16T + 37T^{2} \)
41 \( 1 + 11.2T + 41T^{2} \)
43 \( 1 + 5.83T + 43T^{2} \)
47 \( 1 - 4.23T + 47T^{2} \)
53 \( 1 + 0.524T + 53T^{2} \)
59 \( 1 + 8.12iT - 59T^{2} \)
61 \( 1 - 11.5T + 61T^{2} \)
67 \( 1 + 9.84T + 67T^{2} \)
71 \( 1 - 9.18iT - 71T^{2} \)
73 \( 1 + 8.36T + 73T^{2} \)
79 \( 1 + 8.34T + 79T^{2} \)
83 \( 1 + 6.65iT - 83T^{2} \)
89 \( 1 + 3.31T + 89T^{2} \)
97 \( 1 + 2.18T + 97T^{2} \)
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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

−11.42520601364326069764686644664, −10.48837287385612682672104552445, −9.917383991523556662051525847481, −8.505793681799649610794944601889, −8.323798399051824572076868489633, −7.16218061512296528785624775885, −5.92617006898463412211599565959, −4.69604335267869386351361791235, −3.95829502933816853405886726873, −3.55961340181230839135963913334, 0.05915094869699284443142417404, 1.55678067369191494377496132845, 2.97650293829984345247792035983, 3.97506520982688259638784153862, 5.43568581146771169927755434476, 6.70335024638863435422068616137, 7.50167897237290875887795916817, 8.324270305152705589455915158531, 8.772967816013101916947388478785, 10.42703175055917743575995789504

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