Properties

Label 2-1850-5.4-c1-0-48
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 − 4.37i·7-s i·8-s − 9-s + 2.37·11-s + 2i·12-s − 6.74i·13-s + 4.37·14-s + 16-s + 0.372i·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 − 1.65i·7-s − 0.353i·8-s − 0.333·9-s + 0.715·11-s + 0.577i·12-s − 1.87i·13-s + 1.16·14-s + 0.250·16-s + 0.0902i·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

\(L(1)\) \(\approx\) \(1.575070771\)
\(L(\frac12)\) \(\approx\) \(1.575070771\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.37iT - 7T^{2} \)
11 \( 1 - 2.37T + 11T^{2} \)
13 \( 1 + 6.74iT - 13T^{2} \)
17 \( 1 - 0.372iT - 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 - 4.74iT - 23T^{2} \)
29 \( 1 - 9.11T + 29T^{2} \)
31 \( 1 + 8.37T + 31T^{2} \)
41 \( 1 + 0.372T + 41T^{2} \)
43 \( 1 + 1.62iT - 43T^{2} \)
47 \( 1 - 2.74iT - 47T^{2} \)
53 \( 1 - 4.37iT - 53T^{2} \)
59 \( 1 + 1.25T + 59T^{2} \)
61 \( 1 - 0.372T + 61T^{2} \)
67 \( 1 + 6.74iT - 67T^{2} \)
71 \( 1 - 4.74T + 71T^{2} \)
73 \( 1 - 2.74iT - 73T^{2} \)
79 \( 1 + 6.74T + 79T^{2} \)
83 \( 1 + 10.7iT - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 - 17.1iT - 97T^{2} \)
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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

−8.678679888277732134010284925097, −7.66854360324189428447016652469, −7.59272952095495634395613764824, −6.77966797058911990881403398093, −6.05221953030667607155144760329, −5.08049194951904396995340977804, −4.00314964155926738473941203724, −3.14596098301619792561145915513, −1.38623794677194177544041518889, −0.64101568177406071564906113649, 1.69980309501955514649945022663, 2.65032124187323802949963671772, 3.71203302460700589866617775159, 4.49652811124892316417028724592, 5.14359705671449803779097246134, 6.14338762311085974003707036347, 7.00642997768412131587643184031, 8.584767947445242419790507589658, 8.882457505763335356096428870909, 9.514806772673091842066333107225

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