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Properties

Label 2-1785-5.4-c1-0-49
Degree $2$
Conductor $1785$
Sign $-0.397 - 0.917i$
Analytic cond. $14.2532$
Root an. cond. $3.77535$
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  + 2.75i·2-s + i·3-s − 5.61·4-s + (−2.05 + 0.889i)5-s − 2.75·6-s + i·7-s − 9.96i·8-s − 9-s + (−2.45 − 5.66i)10-s + 5.12·11-s − 5.61i·12-s − 3.06i·13-s − 2.75·14-s + (−0.889 − 2.05i)15-s + 16.2·16-s + i·17-s + ⋯
L(s)  = 1  + 1.95i·2-s + 0.577i·3-s − 2.80·4-s + (−0.917 + 0.397i)5-s − 1.12·6-s + 0.377i·7-s − 3.52i·8-s − 0.333·9-s + (−0.775 − 1.79i)10-s + 1.54·11-s − 1.62i·12-s − 0.849i·13-s − 0.737·14-s + (−0.229 − 0.529i)15-s + 4.07·16-s + 0.242i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1785 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.397 - 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1785 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.397 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1785\)    =    \(3 \cdot 5 \cdot 7 \cdot 17\)
Sign: $-0.397 - 0.917i$
Analytic conductor: \(14.2532\)
Root analytic conductor: \(3.77535\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1785} (1429, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1785,\ (\ :1/2),\ -0.397 - 0.917i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.049593169\)
\(L(\frac12)\) \(\approx\) \(1.049593169\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
5 \( 1 + (2.05 - 0.889i)T \)
7 \( 1 - iT \)
17 \( 1 - iT \)
good2 \( 1 - 2.75iT - 2T^{2} \)
11 \( 1 - 5.12T + 11T^{2} \)
13 \( 1 + 3.06iT - 13T^{2} \)
19 \( 1 - 4.71T + 19T^{2} \)
23 \( 1 + 9.27iT - 23T^{2} \)
29 \( 1 - 0.219T + 29T^{2} \)
31 \( 1 - 1.84T + 31T^{2} \)
37 \( 1 + 9.19iT - 37T^{2} \)
41 \( 1 + 8.15T + 41T^{2} \)
43 \( 1 - 4.99iT - 43T^{2} \)
47 \( 1 + 9.93iT - 47T^{2} \)
53 \( 1 + 2.34iT - 53T^{2} \)
59 \( 1 + 4.05T + 59T^{2} \)
61 \( 1 - 4.39T + 61T^{2} \)
67 \( 1 + 7.20iT - 67T^{2} \)
71 \( 1 + 2.47T + 71T^{2} \)
73 \( 1 - 3.44iT - 73T^{2} \)
79 \( 1 - 11.4T + 79T^{2} \)
83 \( 1 + 8.57iT - 83T^{2} \)
89 \( 1 - 0.890T + 89T^{2} \)
97 \( 1 - 0.845iT - 97T^{2} \)
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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

−9.168832164383297242581978916645, −8.493949662478052621655804399457, −7.996321342309425180694870711417, −6.99974489747021083596913314993, −6.50853613458764760614753448691, −5.63378711574212705797694332485, −4.76591998305137046350254865013, −4.01425030343704692991711906992, −3.27975829481015885559435281677, −0.52996619752673975603179502727, 1.08687042286927088933088466555, 1.56440825876175274764540349901, 3.11584586218547652920908827461, 3.73680279227698435194480665039, 4.47714982337864102126765923883, 5.37659661357323649551494784319, 6.79465754974739492846282807167, 7.73635739615574185318189698191, 8.541980651224765921184048424862, 9.366635219797403419037652680796

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