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Properties

Label 2-805-161.160-c1-0-24
Degree $2$
Conductor $805$
Sign $0.997 + 0.0773i$
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  − 2.54·2-s − 0.210i·3-s + 4.46·4-s − 5-s + 0.535i·6-s + (1.16 + 2.37i)7-s − 6.26·8-s + 2.95·9-s + 2.54·10-s − 0.145i·11-s − 0.940i·12-s − 5.36i·13-s + (−2.94 − 6.04i)14-s + 0.210i·15-s + 7.00·16-s − 0.400·17-s + ⋯
L(s)  = 1  − 1.79·2-s − 0.121i·3-s + 2.23·4-s − 0.447·5-s + 0.218i·6-s + (0.438 + 0.898i)7-s − 2.21·8-s + 0.985·9-s + 0.804·10-s − 0.0437i·11-s − 0.271i·12-s − 1.48i·13-s + (−0.788 − 1.61i)14-s + 0.0543i·15-s + 1.75·16-s − 0.0972·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0773i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(805\)    =    \(5 \cdot 7 \cdot 23\)
Sign: $0.997 + 0.0773i$
Analytic conductor: \(6.42795\)
Root analytic conductor: \(2.53534\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{805} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 805,\ (\ :1/2),\ 0.997 + 0.0773i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.698393 - 0.0270429i\)
\(L(\frac12)\) \(\approx\) \(0.698393 - 0.0270429i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + T \)
7 \( 1 + (-1.16 - 2.37i)T \)
23 \( 1 + (2.42 - 4.13i)T \)
good2 \( 1 + 2.54T + 2T^{2} \)
3 \( 1 + 0.210iT - 3T^{2} \)
11 \( 1 + 0.145iT - 11T^{2} \)
13 \( 1 + 5.36iT - 13T^{2} \)
17 \( 1 + 0.400T + 17T^{2} \)
19 \( 1 - 5.62T + 19T^{2} \)
29 \( 1 + 0.412T + 29T^{2} \)
31 \( 1 + 3.47iT - 31T^{2} \)
37 \( 1 + 3.91iT - 37T^{2} \)
41 \( 1 - 4.91iT - 41T^{2} \)
43 \( 1 - 0.866iT - 43T^{2} \)
47 \( 1 - 9.11iT - 47T^{2} \)
53 \( 1 + 9.61iT - 53T^{2} \)
59 \( 1 + 14.8iT - 59T^{2} \)
61 \( 1 - 8.98T + 61T^{2} \)
67 \( 1 - 14.4iT - 67T^{2} \)
71 \( 1 - 12.7T + 71T^{2} \)
73 \( 1 + 8.71iT - 73T^{2} \)
79 \( 1 - 9.15iT - 79T^{2} \)
83 \( 1 - 11.2T + 83T^{2} \)
89 \( 1 - 3.44T + 89T^{2} \)
97 \( 1 - 13.7T + 97T^{2} \)
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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

−9.851327755243784255583140405023, −9.592853981127289317238637294548, −8.424029006539034975474487544088, −7.84746938346102335133044095016, −7.33669302575592389670189221092, −6.15981382806962131065148743623, −5.12280224255164857506652825309, −3.37847272907819177862436494705, −2.13020308999191434642853013078, −0.896409980774799715983799300513, 0.959624577614664844565227219737, 2.01991754466516845868818674535, 3.73855573995420145324122504370, 4.76833470161334350044346739300, 6.54767909140847633292864329266, 7.16501974237637589255429765583, 7.70922230051181943978108338303, 8.678820988124418556721250526967, 9.408591963434690154842081502030, 10.21560572171588189965185042692

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