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Properties

Label 2-507-507.227-c1-0-2
Degree $2$
Conductor $507$
Sign $-0.652 + 0.757i$
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  + (−0.481 + 1.66i)3-s + (−0.160 + 1.99i)4-s + (−3.64 − 1.46i)7-s + (−2.53 − 1.60i)9-s + (−3.23 − 1.22i)12-s + (−2.59 − 2.5i)13-s + (−3.94 − 0.641i)16-s + (5.49 + 1.47i)19-s + (4.19 − 5.35i)21-s + (−3.31 + 3.74i)25-s + (3.88 − 3.44i)27-s + (3.51 − 7.03i)28-s + (0.438 − 7.24i)31-s + (3.60 − 4.79i)36-s + (−6.69 + 10.1i)37-s + ⋯
L(s)  = 1  + (−0.278 + 0.960i)3-s + (−0.0804 + 0.996i)4-s + (−1.37 − 0.554i)7-s + (−0.845 − 0.534i)9-s + (−0.935 − 0.354i)12-s + (−0.720 − 0.693i)13-s + (−0.987 − 0.160i)16-s + (1.26 + 0.337i)19-s + (0.915 − 1.16i)21-s + (−0.663 + 0.748i)25-s + (0.748 − 0.663i)27-s + (0.663 − 1.32i)28-s + (0.0786 − 1.30i)31-s + (0.600 − 0.799i)36-s + (−1.10 + 1.66i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.652 + 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.652 + 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.652 + 0.757i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.652 + 0.757i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0698747 - 0.152459i\)
\(L(\frac12)\) \(\approx\) \(0.0698747 - 0.152459i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.481 - 1.66i)T \)
13 \( 1 + (2.59 + 2.5i)T \)
good2 \( 1 + (0.160 - 1.99i)T^{2} \)
5 \( 1 + (3.31 - 3.74i)T^{2} \)
7 \( 1 + (3.64 + 1.46i)T + (5.04 + 4.84i)T^{2} \)
11 \( 1 + (-9.93 - 4.71i)T^{2} \)
17 \( 1 + (11.7 - 12.2i)T^{2} \)
19 \( 1 + (-5.49 - 1.47i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (28.9 + 2.33i)T^{2} \)
31 \( 1 + (-0.438 + 7.24i)T + (-30.7 - 3.73i)T^{2} \)
37 \( 1 + (6.69 - 10.1i)T + (-14.5 - 34.0i)T^{2} \)
41 \( 1 + (21.9 - 34.6i)T^{2} \)
43 \( 1 + (12.4 + 2.53i)T + (39.5 + 16.8i)T^{2} \)
47 \( 1 + (43.9 + 16.6i)T^{2} \)
53 \( 1 + (51.4 + 12.6i)T^{2} \)
59 \( 1 + (-18.6 - 55.9i)T^{2} \)
61 \( 1 + (12.4 - 9.38i)T + (16.9 - 58.5i)T^{2} \)
67 \( 1 + (6.63 - 7.80i)T + (-10.7 - 66.1i)T^{2} \)
71 \( 1 + (-70.9 - 2.85i)T^{2} \)
73 \( 1 + (1.60 + 5.15i)T + (-60.0 + 41.4i)T^{2} \)
79 \( 1 + (8.45 + 12.2i)T + (-28.0 + 73.8i)T^{2} \)
83 \( 1 + (-38.5 + 73.4i)T^{2} \)
89 \( 1 + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-1.70 - 16.9i)T + (-95.0 + 19.4i)T^{2} \)
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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

−11.65163634753640032956255229513, −10.29680262719255646570341832920, −9.835000635003275579012786671995, −9.016474076465623049154031638502, −7.85668273828177456663919338978, −6.99589469644383244489053920336, −5.85977898410385106289546796707, −4.71098762333120817415232483426, −3.52632418158682961006849401652, −3.06778052177355812319489869608, 0.097337389466126812515423586290, 1.83616184795605944486403103495, 3.08767748715571094108039965021, 4.94322852684127128953648988569, 5.82474195603312589455397281499, 6.61245367402334725176013945411, 7.27568816860675624004811438676, 8.726620807134414795493618051986, 9.509714092026067525382359702299, 10.23432719980748448688014846121

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