Properties

Label 2-1900-95.64-c1-0-26
Degree $2$
Conductor $1900$
Sign $0.198 + 0.980i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.198 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.198 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $0.198 + 0.980i$
Analytic conductor: \(15.1715\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1/2),\ 0.198 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.011104464\)
\(L(\frac12)\) \(\approx\) \(3.011104464\)
\(L(\frac{3}{2})\) 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

−8.882408196427565187886438379808, −8.319092326215861603820479631276, −7.39137326157231773984834684588, −6.93642892461039073201887242957, −6.23598839867054229115330276982, −4.66735123076153607749901671052, −3.83473836477659851878983711562, −3.18944196904042129440665691987, −1.86272089282453797816172552924, −1.04956958593423298981825162783, 1.58689258098948048546239893077, 2.86234447788486866663245201231, 3.20375408548605782681963070250, 4.47442856083546650546176743773, 5.06142727727122008972986291005, 6.34221817077805003970009513637, 6.97193715897988098242182811561, 8.341324749972574596026141723251, 8.641917040610491878845461199024, 9.309513922050384408746175654251

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