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Properties

Label 2-845-65.4-c1-0-55
Degree $2$
Conductor $845$
Sign $-0.577 + 0.816i$
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  + (0.593 − 1.02i)2-s + (−0.298 − 0.172i)3-s + (0.295 + 0.511i)4-s + (1.71 − 1.44i)5-s + (−0.354 + 0.204i)6-s + (−1.01 − 1.75i)7-s + 3.07·8-s + (−1.44 − 2.49i)9-s + (−0.465 − 2.61i)10-s + (−3.36 − 1.94i)11-s − 0.203i·12-s − 2.40·14-s + (−0.759 + 0.135i)15-s + (1.23 − 2.14i)16-s + (−4.71 + 2.72i)17-s − 3.42·18-s + ⋯
L(s)  = 1  + (0.419 − 0.727i)2-s + (−0.172 − 0.0996i)3-s + (0.147 + 0.255i)4-s + (0.764 − 0.644i)5-s + (−0.144 + 0.0836i)6-s + (−0.383 − 0.664i)7-s + 1.08·8-s + (−0.480 − 0.831i)9-s + (−0.147 − 0.826i)10-s + (−1.01 − 0.585i)11-s − 0.0588i·12-s − 0.644·14-s + (−0.196 + 0.0349i)15-s + (0.308 − 0.535i)16-s + (−1.14 + 0.660i)17-s − 0.806·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.577 + 0.816i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (654, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.577 + 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.883456 - 1.70663i\)
\(L(\frac12)\) \(\approx\) \(0.883456 - 1.70663i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

−10.14354378777755558523000734465, −9.138615203183190288021078419953, −8.327849958960629473888260758761, −7.25719186801629088481785028090, −6.32383738481845811120904247700, −5.36819539256087640251929870007, −4.34645185610491611903862325042, −3.29208845451175183588980496767, −2.33875403535905955255269498699, −0.813525571106033106564311433337, 2.03569893747746902898076858988, 2.83727078692304396939891340409, 4.63426520906854818985046477575, 5.53331109547874190235856122343, 5.84443974246887849217360844142, 7.04263343563615627698876428199, 7.52549011196863430195026395136, 8.800585857050253654073221257435, 9.791824229250929644087211658454, 10.49680386160873893716701140845

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