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Properties

Label 2-1900-19.7-c1-0-7
Degree $2$
Conductor $1900$
Sign $0.998 + 0.0577i$
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  + (−1.60 − 2.77i)3-s + 2.20·7-s + (−3.63 + 6.29i)9-s − 1.20·11-s + (−0.5 + 0.866i)13-s + (−1.07 − 1.85i)17-s + (4.30 − 0.673i)19-s + (−3.53 − 6.11i)21-s + (−4.63 + 8.02i)23-s + 13.6·27-s + (−4.16 + 7.21i)29-s + 8.26·31-s + (1.92 + 3.34i)33-s + 2.20·37-s + 3.20·39-s + ⋯
L(s)  = 1  + (−0.925 − 1.60i)3-s + 0.833·7-s + (−1.21 + 2.09i)9-s − 0.363·11-s + (−0.138 + 0.240i)13-s + (−0.259 − 0.449i)17-s + (0.988 − 0.154i)19-s + (−0.770 − 1.33i)21-s + (−0.966 + 1.67i)23-s + 2.63·27-s + (−0.773 + 1.33i)29-s + 1.48·31-s + (0.335 + 0.581i)33-s + 0.362·37-s + 0.513·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $0.998 + 0.0577i$
Analytic conductor: \(15.1715\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1/2),\ 0.998 + 0.0577i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.025973375\)
\(L(\frac12)\) \(\approx\) \(1.025973375\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.30 + 0.673i)T \)
good3 \( 1 + (1.60 + 2.77i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 - 2.20T + 7T^{2} \)
11 \( 1 + 1.20T + 11T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.07 + 1.85i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (4.63 - 8.02i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.16 - 7.21i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.26T + 31T^{2} \)
37 \( 1 - 2.20T + 37T^{2} \)
41 \( 1 + (-3.30 - 5.72i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.17 - 2.03i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.16 - 7.21i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.134 + 0.232i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.16 - 7.21i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.70 + 2.95i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.06 - 1.84i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.23 - 9.06i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.42 + 4.20i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.20 + 14.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 5.73T + 83T^{2} \)
89 \( 1 + (-5.40 + 9.36i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.87 - 13.6i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.086824403758762505873848910473, −7.909182029818406129137049776541, −7.70680091048384755324084459173, −6.91750653404957084750516921561, −6.03843726188399224183623226201, −5.35513228952178349485362807369, −4.64396280856836566202582769521, −2.99406911376347513375127293788, −1.85807259979956863976177901161, −1.08100977475429113359944964388, 0.49796190962253559380696424579, 2.39847997478822552771687238749, 3.70537306624903219563890008112, 4.41062323851677198724631107512, 5.07178066707239538961796774586, 5.79820073932056835599250064152, 6.53864613135677732812256579750, 7.889329480678175279367112018883, 8.493411680097526226132404418958, 9.487441716893758525199826254941

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