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Properties

Label 2-1900-95.64-c1-0-15
Degree $2$
Conductor $1900$
Sign $0.992 - 0.119i$
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  + (2.18 − 1.26i)3-s + 2.72i·7-s + (1.67 − 2.90i)9-s + 3.31·11-s + (2.81 + 1.62i)13-s + (−2.03 + 1.17i)17-s + (3.11 + 3.04i)19-s + (3.43 + 5.95i)21-s + (1.86 + 1.07i)23-s − 0.893i·27-s + (−1.96 + 3.40i)29-s − 10.1·31-s + (7.23 − 4.17i)33-s − 3.68i·37-s + 8.18·39-s + ⋯
L(s)  = 1  + (1.26 − 0.727i)3-s + 1.03i·7-s + (0.559 − 0.968i)9-s + 0.999·11-s + (0.780 + 0.450i)13-s + (−0.494 + 0.285i)17-s + (0.714 + 0.699i)19-s + (0.750 + 1.29i)21-s + (0.387 + 0.223i)23-s − 0.171i·27-s + (−0.365 + 0.632i)29-s − 1.83·31-s + (1.25 − 0.727i)33-s − 0.605i·37-s + 1.31·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.119i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.119i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $0.992 - 0.119i$
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: \((2,\ 1900,\ (\ :1/2),\ 0.992 - 0.119i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.960786033\)
\(L(\frac12)\) \(\approx\) \(2.960786033\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.11 - 3.04i)T \)
good3 \( 1 + (-2.18 + 1.26i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 - 2.72iT - 7T^{2} \)
11 \( 1 - 3.31T + 11T^{2} \)
13 \( 1 + (-2.81 - 1.62i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.03 - 1.17i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.86 - 1.07i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.96 - 3.40i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 10.1T + 31T^{2} \)
37 \( 1 + 3.68iT - 37T^{2} \)
41 \( 1 + (-0.363 - 0.629i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.05 + 1.18i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-9.54 - 5.51i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.77 + 4.49i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.48 + 9.50i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.22 + 7.32i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.44 - 4.87i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.45 + 5.99i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.14 + 1.24i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.99 - 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.68iT - 83T^{2} \)
89 \( 1 + (-4.27 + 7.39i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.26 + 3.61i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.194445430116946723828294003565, −8.560236627061505048570966065012, −7.75418814113899600793911780057, −6.99799728189840412789725345672, −6.18187687651847937605439597110, −5.32769687087150661209058233289, −3.89599918235711719964442017759, −3.30164139307115497951961367117, −2.12652217394581903139870247448, −1.48618215502878343850189298177, 1.03771176357128095898027406578, 2.45341668694006379055745342282, 3.55317317133323450801466220926, 3.92734775203155822308698213792, 4.84326385125934121974387470344, 6.05043780391933456652879493242, 7.11214314410893925579205247995, 7.63916136755870422135804316722, 8.692570363385894689691025744139, 9.116596789315121832524138624841

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