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Properties

Label 2-506-11.3-c1-0-6
Degree $2$
Conductor $506$
Sign $0.991 - 0.133i$
Analytic cond. $4.04043$
Root an. cond. $2.01008$
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  + (−0.809 + 0.587i)2-s + (−0.348 − 1.07i)3-s + (0.309 − 0.951i)4-s + (2.43 + 1.76i)5-s + (0.913 + 0.663i)6-s + (0.436 − 1.34i)7-s + (0.309 + 0.951i)8-s + (1.39 − 1.01i)9-s − 3.00·10-s + (−0.516 + 3.27i)11-s − 1.12·12-s + (0.705 − 0.512i)13-s + (0.436 + 1.34i)14-s + (1.04 − 3.22i)15-s + (−0.809 − 0.587i)16-s + (1.93 + 1.40i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (−0.201 − 0.620i)3-s + (0.154 − 0.475i)4-s + (1.08 + 0.789i)5-s + (0.373 + 0.271i)6-s + (0.164 − 0.507i)7-s + (0.109 + 0.336i)8-s + (0.465 − 0.337i)9-s − 0.949·10-s + (−0.155 + 0.987i)11-s − 0.326·12-s + (0.195 − 0.142i)13-s + (0.116 + 0.358i)14-s + (0.270 − 0.833i)15-s + (−0.202 − 0.146i)16-s + (0.470 + 0.341i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 506 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.133i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 506 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.133i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(506\)    =    \(2 \cdot 11 \cdot 23\)
Sign: $0.991 - 0.133i$
Analytic conductor: \(4.04043\)
Root analytic conductor: \(2.01008\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{506} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 506,\ (\ :1/2),\ 0.991 - 0.133i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30558 + 0.0875525i\)
\(L(\frac12)\) \(\approx\) \(1.30558 + 0.0875525i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 + (0.516 - 3.27i)T \)
23 \( 1 + T \)
good3 \( 1 + (0.348 + 1.07i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-2.43 - 1.76i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.436 + 1.34i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-0.705 + 0.512i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.93 - 1.40i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.237 + 0.731i)T + (-15.3 + 11.1i)T^{2} \)
29 \( 1 + (0.719 - 2.21i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-7.52 + 5.46i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.873 + 2.68i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-3.09 - 9.51i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 6.13T + 43T^{2} \)
47 \( 1 + (-0.788 - 2.42i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-9.60 + 6.97i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.83 + 5.65i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (6.29 + 4.57i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 13.4T + 67T^{2} \)
71 \( 1 + (-0.244 - 0.177i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.53 - 4.72i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (10.1 - 7.39i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (9.60 + 6.98i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 7.28T + 89T^{2} \)
97 \( 1 + (-2.90 + 2.11i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.60772553015961894353386234119, −10.00871000070096985733964130933, −9.388361342672488952404386106376, −7.956144683509898538966405027860, −7.23540303647878795522685975444, −6.47703135704190480394708165624, −5.77472575456221336331898072518, −4.31066287109288271393340280482, −2.49330593752856296539068020568, −1.30699473663957718724363280747, 1.28918878625170273750312949955, 2.63732283124368282373936568899, 4.16878201697149527206466813929, 5.31535885735067433399708647885, 5.98908775100551858369011796784, 7.48091951845417423548274424259, 8.671397527951576520392854962942, 9.097660653240588757912255866441, 10.12805221619578494633687419909, 10.54503009374530459938146590183

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