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Properties

Label 2-2850-5.4-c1-0-14
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 + 2i·7-s − i·8-s − 9-s − 4·11-s + i·12-s − 6i·13-s − 2·14-s + 16-s + 4i·17-s − i·18-s − 19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s + 0.755i·7-s − 0.353i·8-s − 0.333·9-s − 1.20·11-s + 0.288i·12-s − 1.66i·13-s − 0.534·14-s + 0.250·16-s + 0.970i·17-s − 0.235i·18-s − 0.229·19-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

\(L(1)\) \(\approx\) \(1.336012501\)
\(L(\frac12)\) \(\approx\) \(1.336012501\)
\(L(\frac{3}{2})\) 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 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 - 4iT - 17T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 10T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 - 8T + 41T^{2} \)
43 \( 1 - 8iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 2T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 - 10iT - 83T^{2} \)
89 \( 1 - 12T + 89T^{2} \)
97 \( 1 + 2iT - 97T^{2} \)
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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

−8.486619549428608830459328639220, −8.148408168919827678960818082060, −7.57259251747135335320271882253, −6.58585457551800981860114958620, −5.76276361100783729090954801250, −5.44507793456338685945567395811, −4.40798568885981061719820156992, −3.08900901967576802620184400120, −2.42389629734007886126083555974, −0.860885018401932602538821588881, 0.57454389415014413932229875682, 2.06068917090482858792635915136, 2.91447135895081098722340698801, 3.92844086439447782908691440136, 4.61302521802055308166902626509, 5.19433612099057902841209979553, 6.38378191956660741830353379255, 7.20000570504533894710831757201, 8.009652851944364541376584549456, 8.919522091086955405478468955420

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