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Properties

Label 2-300-60.47-c2-0-2
Degree $2$
Conductor $300$
Sign $-0.672 - 0.740i$
Analytic cond. $8.17440$
Root an. cond. $2.85909$
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  + (1.01 − 1.72i)2-s + (−2.12 + 2.12i)3-s + (−1.93 − 3.5i)4-s + (1.49 + 5.80i)6-s + (−7.99 − 0.218i)8-s − 8.99i·9-s + (11.5 + 3.31i)12-s + (−8.50 + 13.5i)16-s + (−21.9 + 21.9i)17-s + (−15.5 − 9.14i)18-s − 30.9·19-s + (−24.0 + 24.0i)23-s + (17.4 − 16.4i)24-s + (19.0 + 19.0i)27-s − 61.9i·31-s + (14.7 + 28.4i)32-s + ⋯
L(s)  = 1  + (0.507 − 0.861i)2-s + (−0.707 + 0.707i)3-s + (−0.484 − 0.875i)4-s + (0.249 + 0.968i)6-s + (−0.999 − 0.0273i)8-s − 0.999i·9-s + (0.961 + 0.276i)12-s + (−0.531 + 0.847i)16-s + (−1.28 + 1.28i)17-s + (−0.861 − 0.507i)18-s − 1.63·19-s + (−1.04 + 1.04i)23-s + (0.726 − 0.687i)24-s + (0.707 + 0.707i)27-s − 1.99i·31-s + (0.460 + 0.887i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.672 - 0.740i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.672 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.672 - 0.740i$
Analytic conductor: \(8.17440\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1),\ -0.672 - 0.740i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0220195 + 0.0497338i\)
\(L(\frac12)\) \(\approx\) \(0.0220195 + 0.0497338i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.01 + 1.72i)T \)
3 \( 1 + (2.12 - 2.12i)T \)
5 \( 1 \)
good7 \( 1 - 49iT^{2} \)
11 \( 1 + 121T^{2} \)
13 \( 1 - 169iT^{2} \)
17 \( 1 + (21.9 - 21.9i)T - 289iT^{2} \)
19 \( 1 + 30.9T + 361T^{2} \)
23 \( 1 + (24.0 - 24.0i)T - 529iT^{2} \)
29 \( 1 + 841T^{2} \)
31 \( 1 + 61.9iT - 961T^{2} \)
37 \( 1 + 1.36e3iT^{2} \)
41 \( 1 - 1.68e3T^{2} \)
43 \( 1 + 1.84e3iT^{2} \)
47 \( 1 + (9.89 + 9.89i)T + 2.20e3iT^{2} \)
53 \( 1 + (43.8 + 43.8i)T + 2.80e3iT^{2} \)
59 \( 1 - 3.48e3T^{2} \)
61 \( 1 - 118T + 3.72e3T^{2} \)
67 \( 1 - 4.48e3iT^{2} \)
71 \( 1 + 5.04e3T^{2} \)
73 \( 1 - 5.32e3iT^{2} \)
79 \( 1 + 123.T + 6.24e3T^{2} \)
83 \( 1 + (-108. + 108. i)T - 6.88e3iT^{2} \)
89 \( 1 + 7.92e3T^{2} \)
97 \( 1 + 9.40e3iT^{2} \)
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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

−11.64046466481868501339175270354, −11.03053628051605277213911420960, −10.25620185603602573466242177994, −9.413387188251607127577979738693, −8.376174576421330895113671827586, −6.44308078568262431540369146697, −5.79053639281404469945386229400, −4.43747858285360837754523299048, −3.83775684583391264404675007900, −2.04502374222831571906696595921, 0.02289069366073654045357072169, 2.43643748932896669754995406389, 4.32098782739045583461543968297, 5.20347658094757952608470028813, 6.48552533597066627357303647751, 6.85423884310101500702603657689, 8.112478527714177599569252405810, 8.890165874891105776228574727903, 10.41771929957395202195768792572, 11.44112908928183147340871914697

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