Properties

Label 2-1900-19.7-c1-0-1
Degree $2$
Conductor $1900$
Sign $-0.296 - 0.954i$
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  + (−0.0695 − 0.120i)3-s − 3.40·7-s + (1.49 − 2.58i)9-s + 0.185·11-s + (−1.17 + 2.03i)13-s + (3.35 + 5.81i)17-s + (−2.14 − 3.79i)19-s + (0.236 + 0.410i)21-s + (−0.463 + 0.803i)23-s − 0.831·27-s + (0.677 − 1.17i)29-s − 9.46·31-s + (−0.0129 − 0.0223i)33-s − 1.62·37-s + 0.327·39-s + ⋯
L(s)  = 1  + (−0.0401 − 0.0695i)3-s − 1.28·7-s + (0.496 − 0.860i)9-s + 0.0559·11-s + (−0.326 + 0.565i)13-s + (0.814 + 1.41i)17-s + (−0.491 − 0.870i)19-s + (0.0516 + 0.0895i)21-s + (−0.0967 + 0.167i)23-s − 0.160·27-s + (0.125 − 0.217i)29-s − 1.69·31-s + (−0.00224 − 0.00389i)33-s − 0.267·37-s + 0.0524·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.296 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.296 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $-0.296 - 0.954i$
Analytic conductor: \(15.1715\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1/2),\ -0.296 - 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7480632697\)
\(L(\frac12)\) \(\approx\) \(0.7480632697\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.14 + 3.79i)T \)
good3 \( 1 + (0.0695 + 0.120i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + 3.40T + 7T^{2} \)
11 \( 1 - 0.185T + 11T^{2} \)
13 \( 1 + (1.17 - 2.03i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.35 - 5.81i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (0.463 - 0.803i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.677 + 1.17i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 9.46T + 31T^{2} \)
37 \( 1 + 1.62T + 37T^{2} \)
41 \( 1 + (-4.29 - 7.44i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.80 - 6.58i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.09 + 7.09i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.70 - 11.6i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.02 - 3.51i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.31 - 10.9i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.62 - 2.81i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.59 - 11.4i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.45 - 4.24i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.98 + 5.16i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.30T + 83T^{2} \)
89 \( 1 + (-1.95 + 3.38i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.25 - 12.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.371633890641171161543838880268, −8.948967309468619612194950408651, −7.76396065023771415566691372855, −6.96430157400400236972485309199, −6.32448589314524394683700585190, −5.68193739994406799270147113966, −4.29330007586109223563378526123, −3.68548455800015398713136986168, −2.68486528679323345912179829383, −1.28474149138904556803795616018, 0.28308233325644593303004767071, 1.98525737789912676768419921692, 3.08122204026303428931669490698, 3.86408885779725159369698518323, 5.05421444811138936697814209496, 5.67811636064477076646831961690, 6.69338655503755617039921523483, 7.42287882703755486960524387165, 8.034272364946529634953724306542, 9.285968043376489066214094808667

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