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Properties

Label 2-507-507.110-c1-0-51
Degree $2$
Conductor $507$
Sign $-0.192 + 0.981i$
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  + (1.46 − 0.925i)3-s + (−0.320 − 1.97i)4-s + (0.0119 − 0.591i)7-s + (1.28 − 2.71i)9-s + (−2.29 − 2.59i)12-s + (−2.59 − 2.5i)13-s + (−3.79 + 1.26i)16-s + (6.99 + 1.87i)19-s + (−0.529 − 0.876i)21-s + (−4.96 − 0.602i)25-s + (−0.626 − 5.15i)27-s + (−1.17 + 0.166i)28-s + (3.95 + 3.09i)31-s + (−5.76 − 1.66i)36-s + (−1.85 − 4.61i)37-s + ⋯
L(s)  = 1  + (0.845 − 0.534i)3-s + (−0.160 − 0.987i)4-s + (0.00449 − 0.223i)7-s + (0.428 − 0.903i)9-s + (−0.663 − 0.748i)12-s + (−0.720 − 0.693i)13-s + (−0.948 + 0.316i)16-s + (1.60 + 0.430i)19-s + (−0.115 − 0.191i)21-s + (−0.992 − 0.120i)25-s + (−0.120 − 0.992i)27-s + (−0.221 + 0.0313i)28-s + (0.710 + 0.556i)31-s + (−0.960 − 0.278i)36-s + (−0.305 − 0.758i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.192 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.192 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.192 + 0.981i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (110, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.192 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09573 - 1.33182i\)
\(L(\frac12)\) \(\approx\) \(1.09573 - 1.33182i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.46 + 0.925i)T \)
13 \( 1 + (2.59 + 2.5i)T \)
good2 \( 1 + (0.320 + 1.97i)T^{2} \)
5 \( 1 + (4.96 + 0.602i)T^{2} \)
7 \( 1 + (-0.0119 + 0.591i)T + (-6.99 - 0.281i)T^{2} \)
11 \( 1 + (8.52 + 6.95i)T^{2} \)
17 \( 1 + (-0.684 + 16.9i)T^{2} \)
19 \( 1 + (-6.99 - 1.87i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-28.6 + 4.65i)T^{2} \)
31 \( 1 + (-3.95 - 3.09i)T + (7.41 + 30.0i)T^{2} \)
37 \( 1 + (1.85 + 4.61i)T + (-26.6 + 25.6i)T^{2} \)
41 \( 1 + (37.0 + 17.5i)T^{2} \)
43 \( 1 + (-0.560 - 1.31i)T + (-29.7 + 31.0i)T^{2} \)
47 \( 1 + (31.1 + 35.1i)T^{2} \)
53 \( 1 + (-46.9 + 24.6i)T^{2} \)
59 \( 1 + (35.4 - 47.1i)T^{2} \)
61 \( 1 + (-4.23 - 14.6i)T + (-51.5 + 32.6i)T^{2} \)
67 \( 1 + (-13.2 + 9.54i)T + (21.2 - 63.5i)T^{2} \)
71 \( 1 + (-5.71 - 70.7i)T^{2} \)
73 \( 1 + (-2.71 + 0.496i)T + (68.2 - 25.8i)T^{2} \)
79 \( 1 + (1.66 + 4.38i)T + (-59.1 + 52.3i)T^{2} \)
83 \( 1 + (68.3 + 47.1i)T^{2} \)
89 \( 1 + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-9.37 - 14.1i)T + (-38.0 + 89.2i)T^{2} \)
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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

−10.34288569427650958505854517531, −9.795047818998377250647736989540, −8.983926511887925163611775596224, −7.87149653021660045211369896362, −7.16339494941248261248621403404, −6.02773550504360815907420144301, −5.06280101217026728123601899849, −3.69787876338521073336589060147, −2.38274101582383088155272422108, −0.994796604304480743637356959895, 2.28722754106062091237425212772, 3.29671148420053327867844340718, 4.27806855800617464261902627952, 5.24603874099730972435908392247, 6.93992450100553945713532645475, 7.70572822738271945185975013554, 8.483164511394783950599019507512, 9.421046608921855142705114430257, 9.886817941890083411931792288550, 11.33707677756666209708715399275

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