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Properties

Label 2-507-13.12-c1-0-24
Degree $2$
Conductor $507$
Sign $-0.832 + 0.554i$
Analytic cond. $4.04841$
Root an. cond. $2.01206$
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  − 1.56i·2-s + 3-s − 0.438·4-s − 3.56i·5-s − 1.56i·6-s − 0.561i·7-s − 2.43i·8-s + 9-s − 5.56·10-s + 2i·11-s − 0.438·12-s − 0.876·14-s − 3.56i·15-s − 4.68·16-s + 1.56·17-s − 1.56i·18-s + ⋯
L(s)  = 1  − 1.10i·2-s + 0.577·3-s − 0.219·4-s − 1.59i·5-s − 0.637i·6-s − 0.212i·7-s − 0.862i·8-s + 0.333·9-s − 1.75·10-s + 0.603i·11-s − 0.126·12-s − 0.234·14-s − 0.919i·15-s − 1.17·16-s + 0.378·17-s − 0.368i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.832 + 0.554i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.832 + 0.554i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.533978 - 1.76361i\)
\(L(\frac12)\) \(\approx\) \(0.533978 - 1.76361i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
13 \( 1 \)
good2 \( 1 + 1.56iT - 2T^{2} \)
5 \( 1 + 3.56iT - 5T^{2} \)
7 \( 1 + 0.561iT - 7T^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
17 \( 1 - 1.56T + 17T^{2} \)
19 \( 1 - 7.12iT - 19T^{2} \)
23 \( 1 + 2T + 23T^{2} \)
29 \( 1 - 6.68T + 29T^{2} \)
31 \( 1 - 2.56iT - 31T^{2} \)
37 \( 1 + 7.56iT - 37T^{2} \)
41 \( 1 + 1.56iT - 41T^{2} \)
43 \( 1 + 4.56T + 43T^{2} \)
47 \( 1 + 8.24iT - 47T^{2} \)
53 \( 1 + 0.684T + 53T^{2} \)
59 \( 1 - 2.87iT - 59T^{2} \)
61 \( 1 - 3.87T + 61T^{2} \)
67 \( 1 - 4.56iT - 67T^{2} \)
71 \( 1 - 14iT - 71T^{2} \)
73 \( 1 - 10.1iT - 73T^{2} \)
79 \( 1 - 5.43T + 79T^{2} \)
83 \( 1 + 0.876iT - 83T^{2} \)
89 \( 1 + 4.87iT - 89T^{2} \)
97 \( 1 + 8.56iT - 97T^{2} \)
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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

−10.26650781659853042463775570992, −9.920560455357190490451219019700, −8.891621546628824006953028615223, −8.166998183559286295961534493775, −7.08233507105965932126947009269, −5.61686205262064651467550139171, −4.40578367136338687039690521227, −3.65340276902335159327803647804, −2.12270262971521429233173637168, −1.11740343403117292257184358055, 2.45797691614406533300204551605, 3.21565128503775535476955878041, 4.83544634177252912789362391645, 6.18899400453717964598201763068, 6.66981485444052679453035203044, 7.55370717598794542260289370714, 8.275909221572394179972882423539, 9.298515898618026516784610363253, 10.41722600514728792391390716884, 11.14793524407878920827792953534

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