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Properties

Label 2-507-13.4-c1-0-26
Degree $2$
Conductor $507$
Sign $-0.543 + 0.839i$
Analytic cond. $4.04841$
Root an. cond. $2.01206$
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  + (0.866 + 0.5i)2-s + (0.5 − 0.866i)3-s + (−0.500 − 0.866i)4-s − 2i·5-s + (0.866 − 0.499i)6-s + (−3.46 + 2i)7-s − 3i·8-s + (−0.499 − 0.866i)9-s + (1 − 1.73i)10-s + (−3.46 − 2i)11-s − 12-s − 3.99·14-s + (−1.73 − i)15-s + (0.500 − 0.866i)16-s + (1 + 1.73i)17-s − 0.999i·18-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.288 − 0.499i)3-s + (−0.250 − 0.433i)4-s − 0.894i·5-s + (0.353 − 0.204i)6-s + (−1.30 + 0.755i)7-s − 1.06i·8-s + (−0.166 − 0.288i)9-s + (0.316 − 0.547i)10-s + (−1.04 − 0.603i)11-s − 0.288·12-s − 1.06·14-s + (−0.447 − 0.258i)15-s + (0.125 − 0.216i)16-s + (0.242 + 0.420i)17-s − 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.543 + 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.543 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.543 + 0.839i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.543 + 0.839i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.605648 - 1.11443i\)
\(L(\frac12)\) \(\approx\) \(0.605648 - 1.11443i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 \)
good2 \( 1 + (-0.866 - 0.5i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + 2iT - 5T^{2} \)
7 \( 1 + (3.46 - 2i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.46 + 2i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5 + 8.66i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4iT - 31T^{2} \)
37 \( 1 + (-1.73 - i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.19 - 3i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (6 + 10.3i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (-10.3 + 6i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.92 + 4i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 + (-1.73 - i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.66 - 5i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.38516686379216812754785710299, −9.620408432861726135263780124585, −8.812968544246956760293060853275, −7.967242877284151867198205796654, −6.60092293847460045573578207566, −5.89703178190644632425152677084, −5.15793714502192657356880132517, −3.84388913374310136613066332041, −2.57776087182548201477794171579, −0.58427601069559211659278518920, 2.77682253026464955026954085100, 3.20042096471468075968969185059, 4.30324775280008398674875661608, 5.32759007511666517854498648748, 6.76475613307462198498448976832, 7.45832661107519511001384848081, 8.589947577738212527438042482678, 9.708459303304471877678323976374, 10.39387809811603231696194845844, 11.03703227081171774483274474939

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