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Properties

Label 2-513-171.49-c1-0-6
Degree $2$
Conductor $513$
Sign $0.236 + 0.971i$
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  − 2.09·2-s + 2.40·4-s + (−1.44 − 2.50i)5-s + (0.116 + 0.201i)7-s − 0.839·8-s + (3.03 + 5.26i)10-s + (1.99 + 3.45i)11-s + 3.83·13-s + (−0.244 − 0.423i)14-s − 3.03·16-s + (−0.0780 + 0.135i)17-s + (−1.94 − 3.89i)19-s + (−3.47 − 6.01i)20-s + (−4.18 − 7.25i)22-s + 0.942·23-s + ⋯
L(s)  = 1  − 1.48·2-s + 1.20·4-s + (−0.647 − 1.12i)5-s + (0.0440 + 0.0762i)7-s − 0.296·8-s + (0.960 + 1.66i)10-s + (0.601 + 1.04i)11-s + 1.06·13-s + (−0.0653 − 0.113i)14-s − 0.759·16-s + (−0.0189 + 0.0328i)17-s + (−0.447 − 0.894i)19-s + (−0.777 − 1.34i)20-s + (−0.892 − 1.54i)22-s + 0.196·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.236 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.236 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(513\)    =    \(3^{3} \cdot 19\)
Sign: $0.236 + 0.971i$
Analytic conductor: \(4.09632\)
Root analytic conductor: \(2.02393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{513} (334, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 513,\ (\ :1/2),\ 0.236 + 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.449407 - 0.353299i\)
\(L(\frac12)\) \(\approx\) \(0.449407 - 0.353299i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.94 + 3.89i)T \)
good2 \( 1 + 2.09T + 2T^{2} \)
5 \( 1 + (1.44 + 2.50i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.116 - 0.201i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.99 - 3.45i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.83T + 13T^{2} \)
17 \( 1 + (0.0780 - 0.135i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 - 0.942T + 23T^{2} \)
29 \( 1 + (-1.62 + 2.82i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.40 + 4.15i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 11.1T + 37T^{2} \)
41 \( 1 + (-0.0537 - 0.0930i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 10.9T + 43T^{2} \)
47 \( 1 + (-3.39 + 5.88i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.03 + 6.98i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.74 + 9.94i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.49 - 4.31i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 - 7.13T + 67T^{2} \)
71 \( 1 + (-3.33 + 5.77i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.38 + 9.32i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 16.2T + 79T^{2} \)
83 \( 1 + (-5.46 - 9.46i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.25 + 2.18i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.08T + 97T^{2} \)
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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

−10.54256481136295042363490052673, −9.543368142810176889534730700215, −8.852261294212141985516476440195, −8.343897230444573741440919666452, −7.39717614474870417704587185152, −6.48208626540568931231195021836, −4.90727165518146558023749405145, −3.99514290828404990991510180151, −1.94830612852712617158758569411, −0.65848579240628831244077245803, 1.25210783277128305681216422995, 3.00459185423835268355303339331, 4.04569036973326748351807898866, 5.98719048335444575233446781749, 6.81956092710218800845522061860, 7.65038834332710455312943493251, 8.530745899326698824773915750314, 9.088797307882629484119645417792, 10.43135024204924278285566481081, 10.78228423344204828745871228399

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