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Properties

Label 2-513-19.6-c1-0-19
Degree $2$
Conductor $513$
Sign $0.499 + 0.866i$
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  + (1.03 − 0.866i)2-s + (−0.0320 + 0.181i)4-s + (−0.152 − 0.866i)5-s + (0.733 − 1.27i)7-s + (1.47 + 2.54i)8-s + (−0.907 − 0.761i)10-s + (−2.61 − 4.52i)11-s + (4.58 − 1.66i)13-s + (−0.343 − 1.94i)14-s + (3.37 + 1.22i)16-s + (5.46 − 4.58i)17-s + (0.819 + 4.28i)19-s + 0.162·20-s + (−6.61 − 2.40i)22-s + (−0.233 + 1.32i)23-s + ⋯
L(s)  = 1  + (0.729 − 0.612i)2-s + (−0.0160 + 0.0909i)4-s + (−0.0682 − 0.387i)5-s + (0.277 − 0.480i)7-s + (0.520 + 0.901i)8-s + (−0.287 − 0.240i)10-s + (−0.787 − 1.36i)11-s + (1.27 − 0.462i)13-s + (−0.0917 − 0.520i)14-s + (0.844 + 0.307i)16-s + (1.32 − 1.11i)17-s + (0.187 + 0.982i)19-s + 0.0363·20-s + (−1.41 − 0.513i)22-s + (−0.0487 + 0.276i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(513\)    =    \(3^{3} \cdot 19\)
Sign: $0.499 + 0.866i$
Analytic conductor: \(4.09632\)
Root analytic conductor: \(2.02393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{513} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 513,\ (\ :1/2),\ 0.499 + 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87312 - 1.08226i\)
\(L(\frac12)\) \(\approx\) \(1.87312 - 1.08226i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.819 - 4.28i)T \)
good2 \( 1 + (-1.03 + 0.866i)T + (0.347 - 1.96i)T^{2} \)
5 \( 1 + (0.152 + 0.866i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (-0.733 + 1.27i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.61 + 4.52i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.58 + 1.66i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-5.46 + 4.58i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (0.233 - 1.32i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-2.17 - 1.82i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (4.17 - 7.23i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 12.0T + 37T^{2} \)
41 \( 1 + (4.81 + 1.75i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (0.594 + 3.37i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (3.05 + 2.56i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (-1.47 + 8.34i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (3.28 - 2.75i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (1.64 - 9.33i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (2.03 + 1.70i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-2.19 - 12.4i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (1.09 + 0.400i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (-5.81 - 2.11i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-0.520 + 0.902i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (11.9 - 4.35i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-8.88 + 7.45i)T + (16.8 - 95.5i)T^{2} \)
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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

−10.77403438070330420514740838437, −10.35597620595325532470439097726, −8.657910117643781689606687550984, −8.263526529490174829893684727063, −7.21484982988079835812426087435, −5.58911781544484662420301116895, −5.12126050814541584508893303818, −3.60739731073556816603895461877, −3.16308597426602827222226504985, −1.24582686628663398213981302153, 1.73482051193092163574183683597, 3.42026942153551367306482626058, 4.58689014282222007068239343899, 5.42898482232600587015887079510, 6.35010289471351395696997221909, 7.20025157096786011768484394776, 8.132059495654751650014459392652, 9.303373848365646684065061070467, 10.27221048374468388423640085544, 10.91740575875998567139958651421

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