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Properties

Label 2-513-171.122-c1-0-17
Degree $2$
Conductor $513$
Sign $0.958 + 0.285i$
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  + 2.59·2-s + 4.72·4-s + (0.664 − 0.383i)5-s + (−1.86 − 3.23i)7-s + 7.07·8-s + (1.72 − 0.995i)10-s + (0.398 − 0.229i)11-s + 5.26i·13-s + (−4.84 − 8.38i)14-s + 8.88·16-s + (−4.62 − 2.67i)17-s + (−0.255 + 4.35i)19-s + (3.14 − 1.81i)20-s + (1.03 − 0.596i)22-s − 0.257i·23-s + ⋯
L(s)  = 1  + 1.83·2-s + 2.36·4-s + (0.297 − 0.171i)5-s + (−0.705 − 1.22i)7-s + 2.49·8-s + (0.545 − 0.314i)10-s + (0.120 − 0.0693i)11-s + 1.45i·13-s + (−1.29 − 2.24i)14-s + 2.22·16-s + (−1.12 − 0.648i)17-s + (−0.0585 + 0.998i)19-s + (0.702 − 0.405i)20-s + (0.220 − 0.127i)22-s − 0.0535i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.285i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 + 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(513\)    =    \(3^{3} \cdot 19\)
Sign: $0.958 + 0.285i$
Analytic conductor: \(4.09632\)
Root analytic conductor: \(2.02393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{513} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 513,\ (\ :1/2),\ 0.958 + 0.285i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.92986 - 0.572220i\)
\(L(\frac12)\) \(\approx\) \(3.92986 - 0.572220i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.255 - 4.35i)T \)
good2 \( 1 - 2.59T + 2T^{2} \)
5 \( 1 + (-0.664 + 0.383i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.86 + 3.23i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.398 + 0.229i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.26iT - 13T^{2} \)
17 \( 1 + (4.62 + 2.67i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + 0.257iT - 23T^{2} \)
29 \( 1 + (-1.13 + 1.97i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.42 - 3.13i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.986iT - 37T^{2} \)
41 \( 1 + (-4.28 - 7.42i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 1.03T + 43T^{2} \)
47 \( 1 + (7.58 + 4.37i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.95 + 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.674 + 1.16i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.86 - 6.69i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + 10.9iT - 67T^{2} \)
71 \( 1 + (-2.76 + 4.79i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.45 + 2.51i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 4.84iT - 79T^{2} \)
83 \( 1 + (-12.3 + 7.13i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.25 - 2.17i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 2.60iT - 97T^{2} \)
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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

−11.23773586606057143126612630366, −10.23270603526230058694177502463, −9.287797029224419730988385558484, −7.68067214471042658109936408041, −6.62821320594220674357724603200, −6.35712586604255455985621304156, −4.89919573733055530000576769954, −4.19196214486410080415002951902, −3.30606982347789316086398069234, −1.85928402105443688047771456338, 2.38290383006813797939311313337, 2.98265001118181685567949380312, 4.28530257048868455805849387396, 5.34914083911124411175401077492, 6.10003507282462473864740013823, 6.67864632278207432066593914806, 8.035323608360336366006471379202, 9.227633497734884555548184783984, 10.41914517410090511035551481385, 11.20101090753301995801962798629

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