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Properties

Label 2-1850-185.184-c1-0-13
Degree $2$
Conductor $1850$
Sign $0.925 + 0.379i$
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  − 2-s − 2.79i·3-s + 4-s + 2.79i·6-s + 2i·7-s − 8-s − 4.79·9-s + 3.79·11-s − 2.79i·12-s − 0.791·13-s − 2i·14-s + 16-s − 1.58·17-s + 4.79·18-s + 7.58i·19-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.61i·3-s + 0.5·4-s + 1.13i·6-s + 0.755i·7-s − 0.353·8-s − 1.59·9-s + 1.14·11-s − 0.805i·12-s − 0.219·13-s − 0.534i·14-s + 0.250·16-s − 0.383·17-s + 1.12·18-s + 1.73i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.379i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.925 + 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1850\)    =    \(2 \cdot 5^{2} \cdot 37\)
Sign: $0.925 + 0.379i$
Analytic conductor: \(14.7723\)
Root analytic conductor: \(3.84347\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1850} (1849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1850,\ (\ :1/2),\ 0.925 + 0.379i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.170373110\)
\(L(\frac12)\) \(\approx\) \(1.170373110\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 \)
37 \( 1 + (-4.58 + 4i)T \)
good3 \( 1 + 2.79iT - 3T^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 - 3.79T + 11T^{2} \)
13 \( 1 + 0.791T + 13T^{2} \)
17 \( 1 + 1.58T + 17T^{2} \)
19 \( 1 - 7.58iT - 19T^{2} \)
23 \( 1 + 0.791T + 23T^{2} \)
29 \( 1 - 0.791iT - 29T^{2} \)
31 \( 1 - 5.37iT - 31T^{2} \)
41 \( 1 - 5.20T + 41T^{2} \)
43 \( 1 - 6T + 43T^{2} \)
47 \( 1 + 1.58iT - 47T^{2} \)
53 \( 1 - 7.58iT - 53T^{2} \)
59 \( 1 - 7.58iT - 59T^{2} \)
61 \( 1 - 8.20iT - 61T^{2} \)
67 \( 1 - 7.37iT - 67T^{2} \)
71 \( 1 - 9.16T + 71T^{2} \)
73 \( 1 - 9.37iT - 73T^{2} \)
79 \( 1 + 12.7iT - 79T^{2} \)
83 \( 1 + 3.16iT - 83T^{2} \)
89 \( 1 + 6iT - 89T^{2} \)
97 \( 1 + 4.41T + 97T^{2} \)
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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

−8.909636330736408005569499493243, −8.442949212420616238779690318084, −7.54460312046825275577073421034, −7.04362078969881860574569743253, −6.06531672968829225796070706277, −5.78256219401102051597919617962, −4.08371924724310328072477987559, −2.76198081108521410386251023640, −1.88006352610429484373420341575, −1.06032323701064936257801246722, 0.66175947919242374200672917513, 2.42227758976712795500673649177, 3.57536874198477899198307311007, 4.29102316022093311404220217825, 4.99036156826574102900076638247, 6.20351070548196750040574029389, 6.93484938840497610792541529439, 7.915907002615979179476398531443, 8.846009756569710157953340094319, 9.548158400802454632716973321656

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