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Properties

Label 2-1870-85.84-c1-0-54
Degree $2$
Conductor $1870$
Sign $0.846 + 0.531i$
Analytic cond. $14.9320$
Root an. cond. $3.86419$
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  + i·2-s − 1.22·3-s − 4-s + (2.17 − 0.514i)5-s − 1.22i·6-s − 1.17·7-s − i·8-s − 1.48·9-s + (0.514 + 2.17i)10-s − i·11-s + 1.22·12-s + 3.65i·13-s − 1.17i·14-s + (−2.67 + 0.633i)15-s + 16-s + (−2.89 − 2.93i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.709·3-s − 0.5·4-s + (0.973 − 0.230i)5-s − 0.501i·6-s − 0.444·7-s − 0.353i·8-s − 0.496·9-s + (0.162 + 0.688i)10-s − 0.301i·11-s + 0.354·12-s + 1.01i·13-s − 0.314i·14-s + (−0.690 + 0.163i)15-s + 0.250·16-s + (−0.701 − 0.712i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 + 0.531i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.846 + 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1870\)    =    \(2 \cdot 5 \cdot 11 \cdot 17\)
Sign: $0.846 + 0.531i$
Analytic conductor: \(14.9320\)
Root analytic conductor: \(3.86419\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1870} (1189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1870,\ (\ :1/2),\ 0.846 + 0.531i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9592922571\)
\(L(\frac12)\) \(\approx\) \(0.9592922571\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 + (-2.17 + 0.514i)T \)
11 \( 1 + iT \)
17 \( 1 + (2.89 + 2.93i)T \)
good3 \( 1 + 1.22T + 3T^{2} \)
7 \( 1 + 1.17T + 7T^{2} \)
13 \( 1 - 3.65iT - 13T^{2} \)
19 \( 1 + 3.65T + 19T^{2} \)
23 \( 1 - 6.60T + 23T^{2} \)
29 \( 1 - 4.45iT - 29T^{2} \)
31 \( 1 + 2.12iT - 31T^{2} \)
37 \( 1 - 3.79T + 37T^{2} \)
41 \( 1 + 6.16iT - 41T^{2} \)
43 \( 1 + 1.05iT - 43T^{2} \)
47 \( 1 + 8.57iT - 47T^{2} \)
53 \( 1 + 9.87iT - 53T^{2} \)
59 \( 1 + 11.5T + 59T^{2} \)
61 \( 1 + 0.835iT - 61T^{2} \)
67 \( 1 - 1.15iT - 67T^{2} \)
71 \( 1 + 14.1iT - 71T^{2} \)
73 \( 1 - 11.8T + 73T^{2} \)
79 \( 1 + 1.81iT - 79T^{2} \)
83 \( 1 + 16.5iT - 83T^{2} \)
89 \( 1 - 2.65T + 89T^{2} \)
97 \( 1 - 17.2T + 97T^{2} \)
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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

−8.925020694508228252374722799956, −8.731704377796882145684125941638, −7.29565017478269546044357727393, −6.48741377982321004877703850894, −6.20294972109662324433059669447, −5.14652922146049546300491269486, −4.70839205881946250118893328606, −3.28717030982394715423094787774, −2.04869372679485015974520044795, −0.43918680272828483816855456308, 1.08733185906147724539834490422, 2.44990508021927127683347944783, 3.11144651569194748623035540010, 4.46844976197718507836251263568, 5.26341853389880060722704588796, 6.13961459936641263035447973302, 6.53543584944166385001340033398, 7.84806535851464816734207726020, 8.769123910815253218252484004988, 9.461671358836269303476931343868

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