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Properties

Label 2-1900-19.18-c2-0-2
Degree $2$
Conductor $1900$
Sign $0.263 - 0.964i$
Analytic cond. $51.7712$
Root an. cond. $7.19522$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.74i·3-s − 9.79·7-s − 5·9-s + 4·11-s − 11.2i·13-s − 19.5·17-s + (5 − 18.3i)19-s + 36.6i·21-s − 9.79·23-s − 14.9i·27-s − 36.6i·29-s + 36.6i·31-s − 14.9i·33-s + 33.6i·37-s − 42·39-s + ⋯
L(s)  = 1  − 1.24i·3-s − 1.39·7-s − 0.555·9-s + 0.363·11-s − 0.863i·13-s − 1.15·17-s + (0.263 − 0.964i)19-s + 1.74i·21-s − 0.425·23-s − 0.554i·27-s − 1.26i·29-s + 1.18i·31-s − 0.453i·33-s + 0.910i·37-s − 1.07·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 - 0.964i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.263 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $0.263 - 0.964i$
Analytic conductor: \(51.7712\)
Root analytic conductor: \(7.19522\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (1101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1),\ 0.263 - 0.964i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.08681115252\)
\(L(\frac12)\) \(\approx\) \(0.08681115252\)
\(L(2)\) 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 \( 1 + (-5 + 18.3i)T \)
good3 \( 1 + 3.74iT - 9T^{2} \)
7 \( 1 + 9.79T + 49T^{2} \)
11 \( 1 - 4T + 121T^{2} \)
13 \( 1 + 11.2iT - 169T^{2} \)
17 \( 1 + 19.5T + 289T^{2} \)
23 \( 1 + 9.79T + 529T^{2} \)
29 \( 1 + 36.6iT - 841T^{2} \)
31 \( 1 - 36.6iT - 961T^{2} \)
37 \( 1 - 33.6iT - 1.36e3T^{2} \)
41 \( 1 + 36.6iT - 1.68e3T^{2} \)
43 \( 1 + 68.5T + 1.84e3T^{2} \)
47 \( 1 - 9.79T + 2.20e3T^{2} \)
53 \( 1 - 56.1iT - 2.80e3T^{2} \)
59 \( 1 - 73.3iT - 3.48e3T^{2} \)
61 \( 1 - 100T + 3.72e3T^{2} \)
67 \( 1 - 11.2iT - 4.48e3T^{2} \)
71 \( 1 - 36.6iT - 5.04e3T^{2} \)
73 \( 1 + 19.5T + 5.32e3T^{2} \)
79 \( 1 + 109. iT - 6.24e3T^{2} \)
83 \( 1 + 29.3T + 6.88e3T^{2} \)
89 \( 1 - 146. iT - 7.92e3T^{2} \)
97 \( 1 - 123. iT - 9.40e3T^{2} \)
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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

−9.090540284094241102227437234975, −8.360926623925369448510955117107, −7.45451955675244566761647486214, −6.67314197364976116622535032726, −6.43073395322092381783298602363, −5.38261454771521391094404404423, −4.15225107781526813745038783942, −3.06190390045238083023835253805, −2.28612035555357132984621514551, −0.957319116968392446389957088097, 0.02566179177811209413519894636, 1.91141159342762503808870248590, 3.26109704006816808844018487537, 3.85147787726749751525558779307, 4.58392839608896344767036712364, 5.61479893756429847183601068491, 6.50441956100096602615952649546, 7.05855272461290219669864923542, 8.386789402862347362407271916381, 9.110378349958317522645713513368

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