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Properties

Label 2-1900-95.49-c1-0-12
Degree $2$
Conductor $1900$
Sign $0.664 + 0.746i$
Analytic cond. $15.1715$
Root an. cond. $3.89507$
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  + (−2.22 − 1.28i)3-s − 3.56i·7-s + (1.79 + 3.11i)9-s + 4.56·11-s + (−0.866 + 0.5i)13-s + (4.96 + 2.86i)17-s + (4.35 − 0.221i)19-s + (−4.58 + 7.93i)21-s + (−4.84 + 2.79i)23-s − 1.53i·27-s + (3.38 + 5.86i)29-s + 4.59·31-s + (−10.1 − 5.86i)33-s − 3.56i·37-s + 2.56·39-s + ⋯
L(s)  = 1  + (−1.28 − 0.741i)3-s − 1.34i·7-s + (0.599 + 1.03i)9-s + 1.37·11-s + (−0.240 + 0.138i)13-s + (1.20 + 0.695i)17-s + (0.998 − 0.0507i)19-s + (−1.00 + 1.73i)21-s + (−1.01 + 0.583i)23-s − 0.296i·27-s + (0.628 + 1.08i)29-s + 0.826·31-s + (−1.76 − 1.02i)33-s − 0.586i·37-s + 0.411·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.664 + 0.746i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.664 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $0.664 + 0.746i$
Analytic conductor: \(15.1715\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1/2),\ 0.664 + 0.746i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.246652438\)
\(L(\frac12)\) \(\approx\) \(1.246652438\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
19 \( 1 + (-4.35 + 0.221i)T \)
good3 \( 1 + (2.22 + 1.28i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + 3.56iT - 7T^{2} \)
11 \( 1 - 4.56T + 11T^{2} \)
13 \( 1 + (0.866 - 0.5i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-4.96 - 2.86i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (4.84 - 2.79i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.38 - 5.86i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.59T + 31T^{2} \)
37 \( 1 + 3.56iT - 37T^{2} \)
41 \( 1 + (5.35 - 9.27i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-9.79 - 5.65i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.86 + 3.38i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.94 - 1.70i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.38 - 5.86i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.06 + 7.04i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.48 + 3.16i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.515 + 0.892i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-11.0 - 6.36i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.43 + 4.21i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.40iT - 83T^{2} \)
89 \( 1 + (-6.13 - 10.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.27 - 1.31i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.289645698113426596749337664562, −7.945233667275518591853255175899, −7.42758591740184701589399621547, −6.63366245574250942053969633868, −6.12494819343800829009205537793, −5.18867803539148166800756219623, −4.21826914311819954352114443159, −3.38249511460576798057836387804, −1.44471952253781392174823723220, −0.931060673336458747354398034377, 0.825335686299070068595831762352, 2.42336055382173869414259779449, 3.64637853776994842169067742646, 4.59058207489195147280807345452, 5.42704168059648798718227126793, 5.91702980611598161726390765710, 6.62506952552264526421190906667, 7.75556884743596127934376681812, 8.732953063721960163153645291097, 9.567178959229187963330123120978

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