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Properties

Label 2-2075-415.414-c0-0-4
Degree $2$
Conductor $2075$
Sign $0.894 + 0.447i$
Analytic cond. $1.03555$
Root an. cond. $1.01762$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s − 4-s − i·7-s − 11-s − i·12-s + 16-s − i·17-s + 21-s − 2i·23-s + i·27-s + i·28-s + 29-s − 31-s − i·33-s − i·37-s + ⋯
L(s)  = 1  + i·3-s − 4-s − i·7-s − 11-s − i·12-s + 16-s − i·17-s + 21-s − 2i·23-s + i·27-s + i·28-s + 29-s − 31-s − i·33-s − i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2075\)    =    \(5^{2} \cdot 83\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(1.03555\)
Root analytic conductor: \(1.01762\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2075} (2074, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2075,\ (\ :0),\ 0.894 + 0.447i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8049380189\)
\(L(\frac12)\) \(\approx\) \(0.8049380189\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
83 \( 1 + iT \)
good2 \( 1 + T^{2} \)
3 \( 1 - iT - T^{2} \)
7 \( 1 + iT - T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + iT - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + 2iT - T^{2} \)
29 \( 1 - T + T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 + iT - T^{2} \)
41 \( 1 - 2T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T + T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.312328051748165913350225780573, −8.682317555115754450309970601158, −7.74026699496180766627068981625, −7.09443461782045555224580739363, −5.83775406445439763663875945040, −4.87142124011129682018245897126, −4.48145978343151210828838819835, −3.73828257870730216737982499467, −2.65256324103377807441533424717, −0.66327351903648991710780402291, 1.30811364860051115012236909213, 2.40344518652310358969492174139, 3.52426368712441345631021542142, 4.61355217970417960926881679123, 5.57357512142432426093796790830, 5.99872874878125704665135354328, 7.21311120372422971005235542440, 7.910248606256538165084733947665, 8.433061244045632472640250575919, 9.288732634816143347393879302426

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