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Properties

Label 2-1900-95.18-c1-0-2
Degree $2$
Conductor $1900$
Sign $-0.842 + 0.538i$
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  + (−0.814 + 0.814i)3-s + (−1.28 + 1.28i)7-s + 1.67i·9-s − 0.814·11-s + (−2.02 + 2.02i)13-s + (−1.28 + 1.28i)17-s + (4.09 + 1.48i)19-s − 2.10i·21-s + (1.75 + 1.75i)23-s + (−3.80 − 3.80i)27-s − 1.12·29-s − 4.96i·31-s + (0.663 − 0.663i)33-s + (−3.80 − 3.80i)37-s − 3.30i·39-s + ⋯
L(s)  = 1  + (−0.470 + 0.470i)3-s + (−0.487 + 0.487i)7-s + 0.557i·9-s − 0.245·11-s + (−0.561 + 0.561i)13-s + (−0.312 + 0.312i)17-s + (0.940 + 0.341i)19-s − 0.458i·21-s + (0.366 + 0.366i)23-s + (−0.732 − 0.732i)27-s − 0.209·29-s − 0.891i·31-s + (0.115 − 0.115i)33-s + (−0.625 − 0.625i)37-s − 0.528i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.842 + 0.538i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.842 + 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $-0.842 + 0.538i$
Analytic conductor: \(15.1715\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1/2),\ -0.842 + 0.538i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2871584647\)
\(L(\frac12)\) \(\approx\) \(0.2871584647\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.09 - 1.48i)T \)
good3 \( 1 + (0.814 - 0.814i)T - 3iT^{2} \)
7 \( 1 + (1.28 - 1.28i)T - 7iT^{2} \)
11 \( 1 + 0.814T + 11T^{2} \)
13 \( 1 + (2.02 - 2.02i)T - 13iT^{2} \)
17 \( 1 + (1.28 - 1.28i)T - 17iT^{2} \)
23 \( 1 + (-1.75 - 1.75i)T + 23iT^{2} \)
29 \( 1 + 1.12T + 29T^{2} \)
31 \( 1 + 4.96iT - 31T^{2} \)
37 \( 1 + (3.80 + 3.80i)T + 37iT^{2} \)
41 \( 1 + 5.22iT - 41T^{2} \)
43 \( 1 + (1.22 + 1.22i)T + 43iT^{2} \)
47 \( 1 + (4.40 - 4.40i)T - 47iT^{2} \)
53 \( 1 + (5.41 - 5.41i)T - 53iT^{2} \)
59 \( 1 - 3.99T + 59T^{2} \)
61 \( 1 + 6.46T + 61T^{2} \)
67 \( 1 + (6.22 + 6.22i)T + 67iT^{2} \)
71 \( 1 + 13.0iT - 71T^{2} \)
73 \( 1 + (0.985 + 0.985i)T + 73iT^{2} \)
79 \( 1 - 6.19T + 79T^{2} \)
83 \( 1 + (-5.56 - 5.56i)T + 83iT^{2} \)
89 \( 1 + 2.10T + 89T^{2} \)
97 \( 1 + (3.65 + 3.65i)T + 97iT^{2} \)
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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

−9.502768851682375985477705583582, −9.237498329901055214024191979450, −8.007513361392700782659549222778, −7.39940900107396156399769913216, −6.36493532936970412670977350737, −5.58434460207086666648270172324, −4.92297469796448504022425182049, −4.00093776119673174238767188046, −2.89690453489017642055446505467, −1.83569700705777405742554266892, 0.11556218155910306361324650147, 1.26500710779500440932323321547, 2.80852084700579260357831154123, 3.57738945464961719949439952165, 4.83759313599172028722550672108, 5.51240009440081319897621131776, 6.62109527054720830152216675100, 6.96039218621347652744443863606, 7.82151659084705293097276023770, 8.772160433065547148323849978678

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