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Properties

Label 2-2070-5.4-c1-0-21
Degree $2$
Conductor $2070$
Sign $0.316 - 0.948i$
Analytic cond. $16.5290$
Root an. cond. $4.06559$
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 − 4-s + (0.707 − 2.12i)5-s + 2i·7-s − i·8-s + (2.12 + 0.707i)10-s + 4.24·11-s − 0.828i·13-s − 2·14-s + 16-s + 6.82i·17-s − 6.24·19-s + (−0.707 + 2.12i)20-s + 4.24i·22-s + i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (0.316 − 0.948i)5-s + 0.755i·7-s − 0.353i·8-s + (0.670 + 0.223i)10-s + 1.27·11-s − 0.229i·13-s − 0.534·14-s + 0.250·16-s + 1.65i·17-s − 1.43·19-s + (−0.158 + 0.474i)20-s + 0.904i·22-s + 0.208i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2070\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.316 - 0.948i$
Analytic conductor: \(16.5290\)
Root analytic conductor: \(4.06559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2070} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2070,\ (\ :1/2),\ 0.316 - 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.795406418\)
\(L(\frac12)\) \(\approx\) \(1.795406418\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
5 \( 1 + (-0.707 + 2.12i)T \)
23 \( 1 - iT \)
good7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 - 4.24T + 11T^{2} \)
13 \( 1 + 0.828iT - 13T^{2} \)
17 \( 1 - 6.82iT - 17T^{2} \)
19 \( 1 + 6.24T + 19T^{2} \)
29 \( 1 - 3.65T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 + 0.585iT - 37T^{2} \)
41 \( 1 - 6.82T + 41T^{2} \)
43 \( 1 - 10.2iT - 43T^{2} \)
47 \( 1 + 0.828iT - 47T^{2} \)
53 \( 1 + 10.5iT - 53T^{2} \)
59 \( 1 - 8.48T + 59T^{2} \)
61 \( 1 + 0.585T + 61T^{2} \)
67 \( 1 - 3.41iT - 67T^{2} \)
71 \( 1 - 5.65T + 71T^{2} \)
73 \( 1 - 7.65iT - 73T^{2} \)
79 \( 1 - 3.65T + 79T^{2} \)
83 \( 1 + 1.41iT - 83T^{2} \)
89 \( 1 - 9.17T + 89T^{2} \)
97 \( 1 - 6iT - 97T^{2} \)
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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

−9.067041115151478384263832741985, −8.410849774560288694443747320913, −8.073354133070733301045638621887, −6.56520799814592376457942129148, −6.24596870132349772215646284552, −5.44112444665774667049133513137, −4.44904087140195915873529428601, −3.85237141153582884596001411539, −2.25360677275411575432899391000, −1.12210723840349955897370210286, 0.76895750318965256275921715865, 2.10918113468811383981700111709, 2.97318003433432858218085999231, 3.99570044710681063594551256301, 4.57958555560960042480570660286, 5.88973481229376519254923396825, 6.75374571470554442842860986713, 7.18956477835526013588411194887, 8.336155582719007915098940068401, 9.225760228984908953303623623949

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