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Properties

Label 2-513-171.49-c1-0-5
Degree $2$
Conductor $513$
Sign $0.998 - 0.0571i$
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  + 0.194·2-s − 1.96·4-s + (−0.952 − 1.65i)5-s + (1.69 + 2.93i)7-s − 0.771·8-s + (−0.185 − 0.321i)10-s + (0.311 + 0.539i)11-s + 3.68·13-s + (0.329 + 0.571i)14-s + 3.77·16-s + (3.04 − 5.27i)17-s + (−1.14 + 4.20i)19-s + (1.86 + 3.23i)20-s + (0.0606 + 0.105i)22-s + 7.84·23-s + ⋯
L(s)  = 1  + 0.137·2-s − 0.981·4-s + (−0.426 − 0.738i)5-s + (0.640 + 1.10i)7-s − 0.272·8-s + (−0.0586 − 0.101i)10-s + (0.0939 + 0.162i)11-s + 1.02·13-s + (0.0881 + 0.152i)14-s + 0.943·16-s + (0.739 − 1.28i)17-s + (−0.262 + 0.964i)19-s + (0.418 + 0.724i)20-s + (0.0129 + 0.0224i)22-s + 1.63·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0571i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0571i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(513\)    =    \(3^{3} \cdot 19\)
Sign: $0.998 - 0.0571i$
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: \((2,\ 513,\ (\ :1/2),\ 0.998 - 0.0571i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28161 + 0.0366733i\)
\(L(\frac12)\) \(\approx\) \(1.28161 + 0.0366733i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.14 - 4.20i)T \)
good2 \( 1 - 0.194T + 2T^{2} \)
5 \( 1 + (0.952 + 1.65i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.69 - 2.93i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.311 - 0.539i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.68T + 13T^{2} \)
17 \( 1 + (-3.04 + 5.27i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 - 7.84T + 23T^{2} \)
29 \( 1 + (0.592 - 1.02i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.910 - 1.57i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.63T + 37T^{2} \)
41 \( 1 + (2.01 + 3.49i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 5.09T + 43T^{2} \)
47 \( 1 + (6.43 - 11.1i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.93 + 3.34i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.25 - 7.36i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.82 - 3.15i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 - 1.04T + 67T^{2} \)
71 \( 1 + (1.56 - 2.70i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.06 + 3.58i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 16.3T + 79T^{2} \)
83 \( 1 + (5.35 + 9.27i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.25 - 9.09i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 14.6T + 97T^{2} \)
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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

−11.06342460054080788017186287448, −9.772482703398504655881501607872, −8.885127513988957604926780932839, −8.523413228479155368729603864095, −7.54436395202463294438092729527, −5.93047202270824010345238997527, −5.14678569021070337494667089250, −4.38572104280385491358231761824, −3.09085472829652699888494667096, −1.15074766640045270399126928207, 1.06559126126200485159372889303, 3.31911476051795255833977541673, 4.03729721686662964037456287085, 5.05107681761892581421281192238, 6.30988855915708463200294303301, 7.35937794499267786065704632122, 8.192914183944739931937187320709, 9.001894566660015592046095563449, 10.17557334971387389124968855900, 10.92149837575985733730702215863

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