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Properties

Label 2-1925-5.4-c1-0-75
Degree $2$
Conductor $1925$
Sign $-0.894 + 0.447i$
Analytic cond. $15.3712$
Root an. cond. $3.92061$
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  − 3i·3-s + 2·4-s + i·7-s − 6·9-s − 11-s − 6i·12-s − 4i·13-s + 4·16-s − 2i·17-s + 6·19-s + 3·21-s − 5i·23-s + 9i·27-s + 2i·28-s − 10·29-s + ⋯
L(s)  = 1  − 1.73i·3-s + 4-s + 0.377i·7-s − 2·9-s − 0.301·11-s − 1.73i·12-s − 1.10i·13-s + 16-s − 0.485i·17-s + 1.37·19-s + 0.654·21-s − 1.04i·23-s + 1.73i·27-s + 0.377i·28-s − 1.85·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(15.3712\)
Root analytic conductor: \(3.92061\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (1849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1925,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.906419955\)
\(L(\frac12)\) \(\approx\) \(1.906419955\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 - iT \)
11 \( 1 + T \)
good2 \( 1 - 2T^{2} \)
3 \( 1 + 3iT - 3T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
23 \( 1 + 5iT - 23T^{2} \)
29 \( 1 + 10T + 29T^{2} \)
31 \( 1 - T + 31T^{2} \)
37 \( 1 - 5iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 3T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 3iT - 67T^{2} \)
71 \( 1 - T + 71T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 + 6T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 - 15T + 89T^{2} \)
97 \( 1 - 5iT - 97T^{2} \)
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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

−8.433864075910853271145598083125, −7.933687419223263835644763005687, −7.18487615123925684279967921165, −6.77490627278760374960936890099, −5.70433360912774462932238304515, −5.37879743690031259035376113222, −3.34114096980740599410579866379, −2.61945981667518846661467950286, −1.79477473021876510147740357461, −0.64822021510728349110196068581, 1.70351975162453443113240976522, 3.05554046606618444282450578023, 3.68262326194871040397243149528, 4.56144295370923090313982094306, 5.49507806113068231234420414411, 6.12906838299387824467076103398, 7.31586526685318243301953395859, 7.87064471512915219941141711160, 9.221816020585616975505949603445, 9.455118620832563332869024368694

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