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Properties

Label 2-1850-5.4-c1-0-40
Degree $2$
Conductor $1850$
Sign $0.447 + 0.894i$
Analytic cond. $14.7723$
Root an. cond. $3.84347$
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 − 2i·3-s − 4-s + 2·6-s + i·7-s − i·8-s − 9-s + 3·11-s + 2i·12-s − 4i·13-s − 14-s + 16-s − 3i·17-s − i·18-s − 2·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.15i·3-s − 0.5·4-s + 0.816·6-s + 0.377i·7-s − 0.353i·8-s − 0.333·9-s + 0.904·11-s + 0.577i·12-s − 1.10i·13-s − 0.267·14-s + 0.250·16-s − 0.727i·17-s − 0.235i·18-s − 0.458·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1850\)    =    \(2 \cdot 5^{2} \cdot 37\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(14.7723\)
Root analytic conductor: \(3.84347\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1850} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1850,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.578749633\)
\(L(\frac12)\) \(\approx\) \(1.578749633\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 \)
37 \( 1 + iT \)
good3 \( 1 + 2iT - 3T^{2} \)
7 \( 1 - iT - 7T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 + 3iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 - 5T + 31T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 + iT - 43T^{2} \)
47 \( 1 + 12iT - 47T^{2} \)
53 \( 1 - 3iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + 16iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 17iT - 97T^{2} \)
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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

−8.899177216290367638252314334019, −8.109686740664073178919541779159, −7.42751653773487178162056300692, −6.86002888593224999857280811535, −6.01244020853163183918690084216, −5.39845382891670283639741923816, −4.23207593060098564990426900551, −3.10192255222434719448342467617, −1.84889183125781260900045379826, −0.65293521367922629127572194155, 1.29159593380292501720843578493, 2.54604595404544582857000315103, 3.89246028242719574943863524280, 4.13073344291865515346216601470, 4.90775290295400081722568089157, 6.15597997467289713825454436392, 6.88678194097256094925194069278, 8.129848952162743451971820262235, 8.950746532704195704092773024545, 9.469795911187362942598550550258

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