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Properties

Label 2-845-65.9-c1-0-52
Degree $2$
Conductor $845$
Sign $-0.944 - 0.328i$
Analytic cond. $6.74735$
Root an. cond. $2.59756$
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  + (−1.02 − 0.593i)2-s + (−0.298 − 0.172i)3-s + (−0.295 − 0.511i)4-s + (−1.44 − 1.71i)5-s + (0.204 + 0.354i)6-s + (1.75 − 1.01i)7-s + 3.07i·8-s + (−1.44 − 2.49i)9-s + (0.465 + 2.61i)10-s + (1.94 − 3.36i)11-s + 0.203i·12-s − 2.40·14-s + (0.135 + 0.759i)15-s + (1.23 − 2.14i)16-s + (4.71 − 2.72i)17-s + 3.42i·18-s + ⋯
L(s)  = 1  + (−0.727 − 0.419i)2-s + (−0.172 − 0.0996i)3-s + (−0.147 − 0.255i)4-s + (−0.644 − 0.764i)5-s + (0.0836 + 0.144i)6-s + (0.664 − 0.383i)7-s + 1.08i·8-s + (−0.480 − 0.831i)9-s + (0.147 + 0.826i)10-s + (0.585 − 1.01i)11-s + 0.0588i·12-s − 0.644·14-s + (0.0349 + 0.196i)15-s + (0.308 − 0.535i)16-s + (1.14 − 0.660i)17-s + 0.806i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.328i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.944 - 0.328i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.944 - 0.328i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.105671 + 0.626080i\)
\(L(\frac12)\) \(\approx\) \(0.105671 + 0.626080i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.44 + 1.71i)T \)
13 \( 1 \)
good2 \( 1 + (1.02 + 0.593i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (0.298 + 0.172i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (-1.75 + 1.01i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.94 + 3.36i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-4.71 + 2.72i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.94 + 5.09i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.298 - 0.172i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.18T + 31T^{2} \)
37 \( 1 + (4.71 + 2.72i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.0902 + 0.156i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.15 - 0.669i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 12.2iT - 47T^{2} \)
53 \( 1 - 2.42iT - 53T^{2} \)
59 \( 1 + (-3.53 - 6.11i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.38 + 5.85i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.81 - 2.20i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.940 + 1.62i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 8.86iT - 73T^{2} \)
79 \( 1 + 11.1T + 79T^{2} \)
83 \( 1 + 7.83iT - 83T^{2} \)
89 \( 1 + (6.12 - 10.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.02 - 2.90i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.496261123312777462081083680905, −8.952449039743486573056421205008, −8.317146835654744974956234785375, −7.44973103165037572752663010532, −6.14707683792876027537377340501, −5.24628163946175340392133426502, −4.32036666856960109178222854003, −3.07185594893092783497102696775, −1.27152632950227101449310940312, −0.46752734226750876001219818429, 1.85499289202151605216464627475, 3.40881508240638380039142739627, 4.30443715982868780708981843666, 5.47506641317067303461244480534, 6.65858446318303592842262168858, 7.41698747965903292622068718111, 8.230419920318349119380433602151, 8.567385850429477886486893540276, 10.00930090104982678308506225798, 10.33064157412804245223616103727

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