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Properties

Label 2-513-171.7-c1-0-9
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: $\chi_{513} (235, \cdot )$
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

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

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