Properties

Degree $2$
Conductor $4100$
Sign $0.447 + 0.894i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.24i·3-s + 0.858i·7-s − 7.49·9-s + 6.20·11-s + 1.41i·13-s + 3.93i·17-s − 3.82·19-s + 2.78·21-s + 3.06i·23-s + 14.5i·27-s + 8.48·29-s + 1.13·31-s − 20.1i·33-s − 8.49i·37-s + 4.58·39-s + ⋯
L(s)  = 1  − 1.87i·3-s + 0.324i·7-s − 2.49·9-s + 1.87·11-s + 0.392i·13-s + 0.953i·17-s − 0.877·19-s + 0.607·21-s + 0.639i·23-s + 2.80i·27-s + 1.57·29-s + 0.203·31-s − 3.50i·33-s − 1.39i·37-s + 0.734·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4100\)    =    \(2^{2} \cdot 5^{2} \cdot 41\)
Sign: $0.447 + 0.894i$
Motivic weight: \(1\)
Character: $\chi_{4100} (1149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4100,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.974790308\)
\(L(\frac12)\) \(\approx\) \(1.974790308\)
\(L(\frac{3}{2})\) 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 \( 1 \)
41 \( 1 + T \)
good3 \( 1 + 3.24iT - 3T^{2} \)
7 \( 1 - 0.858iT - 7T^{2} \)
11 \( 1 - 6.20T + 11T^{2} \)
13 \( 1 - 1.41iT - 13T^{2} \)
17 \( 1 - 3.93iT - 17T^{2} \)
19 \( 1 + 3.82T + 19T^{2} \)
23 \( 1 - 3.06iT - 23T^{2} \)
29 \( 1 - 8.48T + 29T^{2} \)
31 \( 1 - 1.13T + 31T^{2} \)
37 \( 1 + 8.49iT - 37T^{2} \)
43 \( 1 - 5.34iT - 43T^{2} \)
47 \( 1 - 6.65iT - 47T^{2} \)
53 \( 1 - 6.41iT - 53T^{2} \)
59 \( 1 - 3.06T + 59T^{2} \)
61 \( 1 - 7.41T + 61T^{2} \)
67 \( 1 + 1.79iT - 67T^{2} \)
71 \( 1 + 3.02T + 71T^{2} \)
73 \( 1 - 0.632iT - 73T^{2} \)
79 \( 1 - 14.3T + 79T^{2} \)
83 \( 1 + 8.76iT - 83T^{2} \)
89 \( 1 - 7.89T + 89T^{2} \)
97 \( 1 + 9.86iT - 97T^{2} \)
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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

−8.274614000944226937361056173605, −7.46257968241262147575232299926, −6.74603245619807885496443075875, −6.27061038341598966188331386962, −5.81296172464417125806273525468, −4.47061705737293372137325440476, −3.55286021176547737317131479639, −2.43092277511463363102581008465, −1.67108024071376353396434373532, −0.925453176964695748369816735845, 0.74489697905913696173078571311, 2.46173986802052907594294309606, 3.43675698262255137316389845298, 4.02616370614920737556044129258, 4.66233989158606377212149769204, 5.30374247828748539327403738505, 6.38463214035087349630861486760, 6.81203864206503250776208143621, 8.237525198381083670886484650990, 8.716795026881668273685919150788

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