Properties

Label 2-2800-5.4-c1-0-13
Degree $2$
Conductor $2800$
Sign $0.447 - 0.894i$
Analytic cond. $22.3581$
Root an. cond. $4.72843$
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·7-s + 3·9-s − 11-s − 2i·13-s + 4i·17-s − 2·19-s + 5i·23-s − 29-s + 2·31-s + 3i·37-s + 12·41-s + 11i·43-s − 2i·47-s − 49-s − 6i·53-s + ⋯
L(s)  = 1  + 0.377i·7-s + 9-s − 0.301·11-s − 0.554i·13-s + 0.970i·17-s − 0.458·19-s + 1.04i·23-s − 0.185·29-s + 0.359·31-s + 0.493i·37-s + 1.87·41-s + 1.67i·43-s − 0.291i·47-s − 0.142·49-s − 0.824i·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2800\)    =    \(2^{4} \cdot 5^{2} \cdot 7\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(22.3581\)
Root analytic conductor: \(4.72843\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2800} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2800,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.730338566\)
\(L(\frac12)\) \(\approx\) \(1.730338566\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 - iT \)
good3 \( 1 - 3T^{2} \)
11 \( 1 + T + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 - 4iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 - 5iT - 23T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 - 3iT - 37T^{2} \)
41 \( 1 - 12T + 41T^{2} \)
43 \( 1 - 11iT - 43T^{2} \)
47 \( 1 + 2iT - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 10T + 59T^{2} \)
61 \( 1 - 4T + 61T^{2} \)
67 \( 1 + iT - 67T^{2} \)
71 \( 1 - 3T + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 9T + 79T^{2} \)
83 \( 1 + 2iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 14iT - 97T^{2} \)
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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

−8.987580220894569765943456026911, −7.983756260835750712005208834580, −7.64626743882928180275023609940, −6.56093804928038261803541032505, −5.94224177464664418584149948307, −5.02738150763096456837183848231, −4.20888597714119995921158889225, −3.32876956629738001565344348671, −2.22956998807617659933941189387, −1.18700357345533808297043422356, 0.61307819382894208145769915765, 1.90652841912408102382557785599, 2.88356868792995965327967952381, 4.15525470121499606203466314814, 4.51084993773464118217159838039, 5.57794553737165345132575280987, 6.53115139993931420625363737273, 7.19080763430036140943586240738, 7.74662939867041886646242846993, 8.767676850893497549172386042600

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