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Properties

Label 2-507-39.5-c1-0-22
Degree $2$
Conductor $507$
Sign $0.0869 - 0.996i$
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  + (1.06 + 1.06i)2-s + (1.36 + 1.06i)3-s + 0.267i·4-s + (1.06 + 1.06i)5-s + (0.320 + 2.58i)6-s + (−1 − i)7-s + (1.84 − 1.84i)8-s + (0.732 + 2.90i)9-s + 2.26i·10-s + (−2.90 + 2.90i)11-s + (−0.285 + 0.366i)12-s − 2.12i·14-s + (0.320 + 2.58i)15-s + 4.46·16-s + 5.03·17-s + (−2.31 + 3.87i)18-s + ⋯
L(s)  = 1  + (0.752 + 0.752i)2-s + (0.788 + 0.614i)3-s + 0.133i·4-s + (0.476 + 0.476i)5-s + (0.130 + 1.05i)6-s + (−0.377 − 0.377i)7-s + (0.652 − 0.652i)8-s + (0.244 + 0.969i)9-s + 0.717i·10-s + (−0.877 + 0.877i)11-s + (−0.0823 + 0.105i)12-s − 0.569i·14-s + (0.0827 + 0.668i)15-s + 1.11·16-s + 1.22·17-s + (−0.546 + 0.913i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0869 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0869 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.0869 - 0.996i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.0869 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.05970 + 1.88767i\)
\(L(\frac12)\) \(\approx\) \(2.05970 + 1.88767i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.36 - 1.06i)T \)
13 \( 1 \)
good2 \( 1 + (-1.06 - 1.06i)T + 2iT^{2} \)
5 \( 1 + (-1.06 - 1.06i)T + 5iT^{2} \)
7 \( 1 + (1 + i)T + 7iT^{2} \)
11 \( 1 + (2.90 - 2.90i)T - 11iT^{2} \)
17 \( 1 - 5.03T + 17T^{2} \)
19 \( 1 + (2.73 - 2.73i)T - 19iT^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 7.16iT - 29T^{2} \)
31 \( 1 + (-2.46 + 2.46i)T - 31iT^{2} \)
37 \( 1 + (3.83 + 3.83i)T + 37iT^{2} \)
41 \( 1 + (3.97 + 3.97i)T + 41iT^{2} \)
43 \( 1 - 2.19iT - 43T^{2} \)
47 \( 1 + (4.25 - 4.25i)T - 47iT^{2} \)
53 \( 1 + 0.779iT - 53T^{2} \)
59 \( 1 + (-2.12 + 2.12i)T - 59iT^{2} \)
61 \( 1 + 7T + 61T^{2} \)
67 \( 1 + (-4.19 + 4.19i)T - 67iT^{2} \)
71 \( 1 + (2.12 + 2.12i)T + 71iT^{2} \)
73 \( 1 + (-0.901 - 0.901i)T + 73iT^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 + (-2.90 - 2.90i)T + 83iT^{2} \)
89 \( 1 + (6.59 - 6.59i)T - 89iT^{2} \)
97 \( 1 + (-1.19 + 1.19i)T - 97iT^{2} \)
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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

−10.68222051086214007352103715595, −10.06679496142554137007868787342, −9.708506911760412736093445210948, −8.124084452470158057369566999984, −7.46891806471119393734886165262, −6.42998586747290130274468326272, −5.45722402502124114400138241417, −4.49409294684414172859055337584, −3.52456466693029530013093430167, −2.17094398784011755504368872773, 1.51412910589809620798916110067, 2.83687476844592739171193366044, 3.39231434571532344651874479129, 4.91983511466198489683161824109, 5.81364369004241795540756719931, 7.11326567124950802324996379458, 8.211244683281357838616523503402, 8.764778590721564140297596471167, 9.885794444180438478666755455154, 10.83923739966871750847440594163

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