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Properties

Label 2-513-171.106-c1-0-0
Degree $2$
Conductor $513$
Sign $0.223 - 0.974i$
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  + (−0.616 − 1.06i)2-s + (0.239 − 0.414i)4-s + 0.551·5-s + (−1.62 + 2.80i)7-s − 3.05·8-s + (−0.340 − 0.589i)10-s + (−2.68 + 4.65i)11-s + (−1.76 + 3.05i)13-s + 4.00·14-s + (1.40 + 2.43i)16-s + (−2.60 + 4.50i)17-s + (0.164 − 4.35i)19-s + (0.131 − 0.228i)20-s + 6.62·22-s + (1.49 − 2.58i)23-s + ⋯
L(s)  = 1  + (−0.436 − 0.755i)2-s + (0.119 − 0.207i)4-s + 0.246·5-s + (−0.612 + 1.06i)7-s − 1.08·8-s + (−0.107 − 0.186i)10-s + (−0.810 + 1.40i)11-s + (−0.488 + 0.846i)13-s + 1.06·14-s + (0.351 + 0.609i)16-s + (−0.630 + 1.09i)17-s + (0.0376 − 0.999i)19-s + (0.0294 − 0.0510i)20-s + 1.41·22-s + (0.311 − 0.538i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.223 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.223 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(513\)    =    \(3^{3} \cdot 19\)
Sign: $0.223 - 0.974i$
Analytic conductor: \(4.09632\)
Root analytic conductor: \(2.02393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{513} (505, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 513,\ (\ :1/2),\ 0.223 - 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.412270 + 0.328267i\)
\(L(\frac12)\) \(\approx\) \(0.412270 + 0.328267i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.164 + 4.35i)T \)
good2 \( 1 + (0.616 + 1.06i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - 0.551T + 5T^{2} \)
7 \( 1 + (1.62 - 2.80i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.68 - 4.65i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.76 - 3.05i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.60 - 4.50i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.49 + 2.58i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.49T + 29T^{2} \)
31 \( 1 + (2.54 + 4.40i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 9.20T + 37T^{2} \)
41 \( 1 + 1.71T + 41T^{2} \)
43 \( 1 + (-1.79 - 3.11i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 10.6T + 47T^{2} \)
53 \( 1 + (0.562 + 0.973i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 7.77T + 59T^{2} \)
61 \( 1 + 11.3T + 61T^{2} \)
67 \( 1 + (1.18 - 2.05i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.507 + 0.879i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.98 - 10.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.568 - 0.985i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.14 + 1.98i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.12 + 1.94i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.09 + 7.08i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.04667049497640582254268723516, −10.08564861970013756503940327571, −9.431545507737318586754041666363, −8.905476807826476907219926797350, −7.48063910812976935461122485051, −6.40940168590094514132587816693, −5.59025338279456115790330585225, −4.31346942856274838545029149023, −2.53977564984539410728821912737, −2.07573602515773988786268291216, 0.32568155108788472780639668706, 2.80957830861537448684162091683, 3.71495632988252509859072283820, 5.44372025820323206211984471223, 6.15078811870831944709420209962, 7.37739301654029322245856561193, 7.69543425412387603931866968447, 8.816109468656885083784986290510, 9.698760323482503775345617805306, 10.59623937894463414258368693181

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