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Properties

Label 2-1900-95.37-c1-0-24
Degree $2$
Conductor $1900$
Sign $0.314 + 0.949i$
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  + (1.63 + 1.63i)3-s + (−2.17 − 2.17i)7-s + 2.36i·9-s − 1.70·11-s + (−4.43 − 4.43i)13-s + (1.63 + 1.63i)17-s + (2.80 − 3.34i)19-s − 7.11i·21-s + (−3.24 + 3.24i)23-s + (1.03 − 1.03i)27-s + 3.27·29-s − 7.11i·31-s + (−2.80 − 2.80i)33-s + (0.128 − 0.128i)37-s − 14.5i·39-s + ⋯
L(s)  = 1  + (0.945 + 0.945i)3-s + (−0.820 − 0.820i)7-s + 0.789i·9-s − 0.515·11-s + (−1.23 − 1.23i)13-s + (0.395 + 0.395i)17-s + (0.642 − 0.766i)19-s − 1.55i·21-s + (−0.677 + 0.677i)23-s + (0.198 − 0.198i)27-s + 0.608·29-s − 1.27i·31-s + (−0.487 − 0.487i)33-s + (0.0211 − 0.0211i)37-s − 2.32i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.314 + 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.314 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $0.314 + 0.949i$
Analytic conductor: \(15.1715\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (1557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1/2),\ 0.314 + 0.949i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.400320957\)
\(L(\frac12)\) \(\approx\) \(1.400320957\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.80 + 3.34i)T \)
good3 \( 1 + (-1.63 - 1.63i)T + 3iT^{2} \)
7 \( 1 + (2.17 + 2.17i)T + 7iT^{2} \)
11 \( 1 + 1.70T + 11T^{2} \)
13 \( 1 + (4.43 + 4.43i)T + 13iT^{2} \)
17 \( 1 + (-1.63 - 1.63i)T + 17iT^{2} \)
23 \( 1 + (3.24 - 3.24i)T - 23iT^{2} \)
29 \( 1 - 3.27T + 29T^{2} \)
31 \( 1 + 7.11iT - 31T^{2} \)
37 \( 1 + (-0.128 + 0.128i)T - 37iT^{2} \)
41 \( 1 + 3.83iT - 41T^{2} \)
43 \( 1 + (-5.24 + 5.24i)T - 43iT^{2} \)
47 \( 1 + (0.908 + 0.908i)T + 47iT^{2} \)
53 \( 1 + (5.47 + 5.47i)T + 53iT^{2} \)
59 \( 1 + 13.9T + 59T^{2} \)
61 \( 1 - 2.63T + 61T^{2} \)
67 \( 1 + (-7.71 + 7.71i)T - 67iT^{2} \)
71 \( 1 - 1.51iT - 71T^{2} \)
73 \( 1 + (-8.70 + 8.70i)T - 73iT^{2} \)
79 \( 1 - 4.78T + 79T^{2} \)
83 \( 1 + (6.46 - 6.46i)T - 83iT^{2} \)
89 \( 1 + 5.34T + 89T^{2} \)
97 \( 1 + (10.2 - 10.2i)T - 97iT^{2} \)
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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

−9.474421568008688113085549985342, −8.153401216814903816185206697677, −7.73882610549040770816410555883, −6.85981050198373246810356008154, −5.68101554581676603718581219342, −4.85324434122253603274104444770, −3.87990953145821955765189651037, −3.21453035327773209568160231045, −2.45015442152031075964455844435, −0.42801790920419676168375386378, 1.53629259503570844216184496940, 2.60971551672145017262228683724, 3.00661033015019033411592241993, 4.41677879790825434440254210278, 5.43751840243224142487524711772, 6.44907635982688627185045340044, 7.05946498677556137260687946741, 7.86157187279390869401314169347, 8.455400264966895265016242553131, 9.457458451074200406514567156028

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