Properties

Label 2-2850-5.4-c1-0-54
Degree $2$
Conductor $2850$
Sign $-0.447 - 0.894i$
Analytic cond. $22.7573$
Root an. cond. $4.77046$
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 i·3-s − 4-s − 6-s − 4i·7-s + i·8-s − 9-s + i·12-s − 6i·13-s − 4·14-s + 16-s − 2i·17-s + i·18-s − 19-s − 4·21-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s − 1.51i·7-s + 0.353i·8-s − 0.333·9-s + 0.288i·12-s − 1.66i·13-s − 1.06·14-s + 0.250·16-s − 0.485i·17-s + 0.235i·18-s − 0.229·19-s − 0.872·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2850\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 19\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(22.7573\)
Root analytic conductor: \(4.77046\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2850} (799, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2850,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9671607078\)
\(L(\frac12)\) \(\approx\) \(0.9671607078\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
5 \( 1 \)
19 \( 1 + T \)
good7 \( 1 + 4iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
23 \( 1 - 8iT - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + 10iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 2iT - 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 - 16T + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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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

−8.153685694424773513054847193544, −7.40976707091801464701384535056, −7.16643917689858276515281674832, −5.73685866075118849776137662862, −5.26851736532864244408542620485, −3.93355982653213919556634538434, −3.51319819809025100368643252288, −2.36823222183087952222672230080, −1.19556409811313956677762926692, −0.32847670179714550338281588766, 1.84806203903419605008932962558, 2.81561454243333712503913721757, 4.03559650739469845883376899103, 4.68713193681521560981260189493, 5.50364735398736847891208494219, 6.29011011853380053769828032362, 6.75990377403134833462161881900, 7.988183746019224980298770398170, 8.618374181995704999307424050565, 9.185434879203501535050741887237

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