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Properties

Label 2-1900-475.284-c0-0-0
Degree $2$
Conductor $1900$
Sign $0.604 - 0.796i$
Analytic cond. $0.948223$
Root an. cond. $0.973767$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.913 + 0.406i)5-s + 1.98i·7-s + (0.809 − 0.587i)9-s + (−0.169 − 0.122i)11-s + (−0.395 − 0.128i)17-s + (0.309 − 0.951i)19-s + (−0.690 + 0.951i)23-s + (0.669 + 0.743i)25-s + (−0.809 + 1.81i)35-s − 1.48i·43-s + (0.978 − 0.207i)45-s + (−1.64 + 0.535i)47-s − 2.95·49-s + (−0.104 − 0.181i)55-s + (1.08 + 0.786i)61-s + ⋯
L(s)  = 1  + (0.913 + 0.406i)5-s + 1.98i·7-s + (0.809 − 0.587i)9-s + (−0.169 − 0.122i)11-s + (−0.395 − 0.128i)17-s + (0.309 − 0.951i)19-s + (−0.690 + 0.951i)23-s + (0.669 + 0.743i)25-s + (−0.809 + 1.81i)35-s − 1.48i·43-s + (0.978 − 0.207i)45-s + (−1.64 + 0.535i)47-s − 2.95·49-s + (−0.104 − 0.181i)55-s + (1.08 + 0.786i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.604 - 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.604 - 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $0.604 - 0.796i$
Analytic conductor: \(0.948223\)
Root analytic conductor: \(0.973767\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (1709, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :0),\ 0.604 - 0.796i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.387535669\)
\(L(\frac12)\) \(\approx\) \(1.387535669\)
\(L(1)\) 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 \( 1 + (-0.913 - 0.406i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
good3 \( 1 + (-0.809 + 0.587i)T^{2} \)
7 \( 1 - 1.98iT - T^{2} \)
11 \( 1 + (0.169 + 0.122i)T + (0.309 + 0.951i)T^{2} \)
13 \( 1 + (0.309 - 0.951i)T^{2} \)
17 \( 1 + (0.395 + 0.128i)T + (0.809 + 0.587i)T^{2} \)
23 \( 1 + (0.690 - 0.951i)T + (-0.309 - 0.951i)T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 + 1.48iT - T^{2} \)
47 \( 1 + (1.64 - 0.535i)T + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (-1.08 - 0.786i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 + (-0.809 - 0.587i)T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (-1.01 + 1.40i)T + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (-1.80 - 0.587i)T + (0.809 + 0.587i)T^{2} \)
89 \( 1 + (-0.309 - 0.951i)T^{2} \)
97 \( 1 + (-0.809 + 0.587i)T^{2} \)
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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

−9.369448016188834995032508297446, −9.045725006891064127542133818698, −8.064291923546948392177857776462, −6.94370281609777686855636186570, −6.30649034647479319336615199043, −5.55011037478668580521394434174, −4.91772627553717848266137975445, −3.47275764992174622806522570664, −2.53536853596708169386767651223, −1.76318870588164673261242918358, 1.13878848091502781701357781653, 2.07887834805423090400553115313, 3.60735750712422119654983116102, 4.44297722143476406804606136319, 5.03324236945600194057720994437, 6.30036000443577061220709131208, 6.86477582949882531414521372579, 7.78961385148909966339772431587, 8.294656202096446637078867786065, 9.679205557456578132857207063395

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