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Properties

Label 2-805-7.2-c1-0-27
Degree $2$
Conductor $805$
Sign $0.827 - 0.560i$
Analytic cond. $6.42795$
Root an. cond. $2.53534$
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  + (−0.831 + 1.44i)2-s + (0.400 + 0.693i)3-s + (−0.383 − 0.664i)4-s + (0.5 − 0.866i)5-s − 1.33·6-s + (−2.64 + 0.0226i)7-s − 2.05·8-s + (1.17 − 2.04i)9-s + (0.831 + 1.44i)10-s + (−3.13 − 5.43i)11-s + (0.307 − 0.532i)12-s + 5.76·13-s + (2.16 − 3.83i)14-s + 0.800·15-s + (2.47 − 4.28i)16-s + (2.77 + 4.79i)17-s + ⋯
L(s)  = 1  + (−0.588 + 1.01i)2-s + (0.231 + 0.400i)3-s + (−0.191 − 0.332i)4-s + (0.223 − 0.387i)5-s − 0.543·6-s + (−0.999 + 0.00857i)7-s − 0.725·8-s + (0.393 − 0.680i)9-s + (0.263 + 0.455i)10-s + (−0.946 − 1.63i)11-s + (0.0886 − 0.153i)12-s + 1.59·13-s + (0.579 − 1.02i)14-s + 0.206·15-s + (0.618 − 1.07i)16-s + (0.672 + 1.16i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 - 0.560i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 - 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(805\)    =    \(5 \cdot 7 \cdot 23\)
Sign: $0.827 - 0.560i$
Analytic conductor: \(6.42795\)
Root analytic conductor: \(2.53534\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{805} (576, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 805,\ (\ :1/2),\ 0.827 - 0.560i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06520 + 0.326913i\)
\(L(\frac12)\) \(\approx\) \(1.06520 + 0.326913i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (2.64 - 0.0226i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (0.831 - 1.44i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.400 - 0.693i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (3.13 + 5.43i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.76T + 13T^{2} \)
17 \( 1 + (-2.77 - 4.79i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.455 + 0.788i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 - 8.41T + 29T^{2} \)
31 \( 1 + (-1.92 - 3.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.81 + 6.60i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.75T + 41T^{2} \)
43 \( 1 + 0.389T + 43T^{2} \)
47 \( 1 + (-2.45 + 4.26i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.87 + 8.44i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.00 + 3.47i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.80 + 8.32i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.61 + 6.25i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.72T + 71T^{2} \)
73 \( 1 + (-0.0846 - 0.146i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.56 - 13.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9.52T + 83T^{2} \)
89 \( 1 + (2.40 - 4.15i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 17.6T + 97T^{2} \)
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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

−10.10057467634802224492792448089, −9.235580830554613982930052658946, −8.420296946593604661554059264367, −8.200346283537083515091623816642, −6.65273778019771830851021337969, −6.20183458516072383306646911660, −5.43595739605083886862343890979, −3.65751665732164900853021004723, −3.16974019993049446571460026815, −0.75749442270165627178647502711, 1.24918733017849816683022910640, 2.46819152243971667364304171895, 3.10977292710049628944547439408, 4.60164168033522316747009482432, 5.93854038053854554252189378377, 6.81985400775342740016135166700, 7.69955603915029227640456837200, 8.665580412452206036587573819618, 9.754285884482113084359616188500, 10.12715520576189637207111114274

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