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Properties

Label 2-1950-65.4-c1-0-8
Degree $2$
Conductor $1950$
Sign $0.322 - 0.946i$
Analytic cond. $15.5708$
Root an. cond. $3.94598$
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  + (−0.5 + 0.866i)2-s + (−0.866 − 0.5i)3-s + (−0.499 − 0.866i)4-s + (0.866 − 0.499i)6-s + (−1.36 − 2.36i)7-s + 0.999·8-s + (0.499 + 0.866i)9-s + (1.09 + 0.633i)11-s + 0.999i·12-s + (−2.5 − 2.59i)13-s + 2.73·14-s + (−0.5 + 0.866i)16-s + (−4.96 + 2.86i)17-s − 0.999·18-s + (−4.09 + 2.36i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.499 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (0.353 − 0.204i)6-s + (−0.516 − 0.894i)7-s + 0.353·8-s + (0.166 + 0.288i)9-s + (0.331 + 0.191i)11-s + 0.288i·12-s + (−0.693 − 0.720i)13-s + 0.730·14-s + (−0.125 + 0.216i)16-s + (−1.20 + 0.695i)17-s − 0.235·18-s + (−0.940 + 0.542i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.322 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.322 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.322 - 0.946i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 0.322 - 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7093791290\)
\(L(\frac12)\) \(\approx\) \(0.7093791290\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 \)
13 \( 1 + (2.5 + 2.59i)T \)
good7 \( 1 + (1.36 + 2.36i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.09 - 0.633i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (4.96 - 2.86i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.09 - 2.36i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.63 - 2.09i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.23 - 3.86i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.46iT - 31T^{2} \)
37 \( 1 + (-1.76 + 3.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-8.13 - 4.69i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-8.36 + 4.83i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.19T + 47T^{2} \)
53 \( 1 + 6.46iT - 53T^{2} \)
59 \( 1 + (-6.92 + 4i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.59 - 7.96i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.56 - 11.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.09 + 2.36i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 6.26T + 73T^{2} \)
79 \( 1 - 2.53T + 79T^{2} \)
83 \( 1 - 0.196T + 83T^{2} \)
89 \( 1 + (-8.19 - 4.73i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3 - 5.19i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.270260197679168064015209095736, −8.503385585405299099579097963713, −7.53914308378577986470450223806, −7.02888613060085771434738805624, −6.33524267824542268484464267993, −5.53899519571204408873603913648, −4.53621278437423131156441242397, −3.74227391064698335866346748139, −2.22489985473569677579526777026, −0.853426987254315009453191582461, 0.41749332664145819041727744440, 2.16674946656727430478186301230, 2.82443283507904852970974728313, 4.20479579491722006989347951499, 4.72569401628877931993790594251, 5.91083659788404491074606441399, 6.60811824940103760070520816662, 7.42953616562852979021735971664, 8.654594812915839938080870401398, 9.233280514830432980454649274941

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