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Properties

Label 2-538-269.268-c1-0-14
Degree $2$
Conductor $538$
Sign $0.0996 + 0.995i$
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 − 1.53i·3-s − 4-s + 3.51·5-s − 1.53·6-s + 1.46i·7-s + i·8-s + 0.641·9-s − 3.51i·10-s + 2.87·11-s + 1.53i·12-s + 1.66·13-s + 1.46·14-s − 5.40i·15-s + 16-s − 0.593i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.886i·3-s − 0.5·4-s + 1.57·5-s − 0.626·6-s + 0.553i·7-s + 0.353i·8-s + 0.213·9-s − 1.11i·10-s + 0.865·11-s + 0.443i·12-s + 0.460·13-s + 0.391·14-s − 1.39i·15-s + 0.250·16-s − 0.143i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0996 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0996 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(538\)    =    \(2 \cdot 269\)
Sign: $0.0996 + 0.995i$
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: \((2,\ 538,\ (\ :1/2),\ 0.0996 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41291 - 1.27854i\)
\(L(\frac12)\) \(\approx\) \(1.41291 - 1.27854i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-16.3 + 1.63i)T \)
good3 \( 1 + 1.53iT - 3T^{2} \)
5 \( 1 - 3.51T + 5T^{2} \)
7 \( 1 - 1.46iT - 7T^{2} \)
11 \( 1 - 2.87T + 11T^{2} \)
13 \( 1 - 1.66T + 13T^{2} \)
17 \( 1 + 0.593iT - 17T^{2} \)
19 \( 1 - 4.67iT - 19T^{2} \)
23 \( 1 + 3.91T + 23T^{2} \)
29 \( 1 + 6.42iT - 29T^{2} \)
31 \( 1 - 7.26iT - 31T^{2} \)
37 \( 1 + 9.91T + 37T^{2} \)
41 \( 1 + 4.88T + 41T^{2} \)
43 \( 1 + 12.5T + 43T^{2} \)
47 \( 1 + 5.42T + 47T^{2} \)
53 \( 1 - 5.55T + 53T^{2} \)
59 \( 1 + 11.1iT - 59T^{2} \)
61 \( 1 + 4.34T + 61T^{2} \)
67 \( 1 + 6.51T + 67T^{2} \)
71 \( 1 + 8.19iT - 71T^{2} \)
73 \( 1 - 1.03T + 73T^{2} \)
79 \( 1 - 4.28T + 79T^{2} \)
83 \( 1 - 16.7iT - 83T^{2} \)
89 \( 1 - 6.07T + 89T^{2} \)
97 \( 1 - 12.2T + 97T^{2} \)
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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

−10.37938810237226794935334656219, −9.907775375249318918737801875358, −8.985486193702039436270796183492, −8.147643554941743646842793950155, −6.69877519558436893027844618275, −6.11807041532103573916434014428, −5.10010505608381989486651356053, −3.55404846017821729785672213438, −2.03408527579213566819331891315, −1.52660865951433510631966602581, 1.62219150454502880701789499477, 3.50995385223220903134250914850, 4.58070784953288673384181642523, 5.47099923174319052689621031271, 6.44595198047561979362505806061, 7.14524400852132381995797635590, 8.703179281467601625254239522040, 9.249081290331785530193773807641, 10.12018606574832443062530698080, 10.51644031564366691109144307225

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