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Properties

Label 2-300-60.47-c2-0-62
Degree $2$
Conductor $300$
Sign $-0.989 - 0.145i$
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  + (−1.81 + 0.837i)2-s + (1.32 − 2.69i)3-s + (2.59 − 3.04i)4-s + (−0.144 + 5.99i)6-s + (−3.54 + 3.54i)7-s + (−2.16 + 7.70i)8-s + (−5.50 − 7.12i)9-s − 16.8·11-s + (−4.76 − 11.0i)12-s + (8.64 − 8.64i)13-s + (3.46 − 9.40i)14-s + (−2.51 − 15.8i)16-s + (−9.72 + 9.72i)17-s + (15.9 + 8.31i)18-s − 4.78·19-s + ⋯
L(s)  = 1  + (−0.908 + 0.418i)2-s + (0.440 − 0.897i)3-s + (0.649 − 0.760i)4-s + (−0.0241 + 0.999i)6-s + (−0.506 + 0.506i)7-s + (−0.270 + 0.962i)8-s + (−0.611 − 0.791i)9-s − 1.53·11-s + (−0.396 − 0.917i)12-s + (0.665 − 0.665i)13-s + (0.247 − 0.671i)14-s + (−0.157 − 0.987i)16-s + (−0.572 + 0.572i)17-s + (0.886 + 0.462i)18-s − 0.251·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.145i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.989 - 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.989 - 0.145i$
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: \((2,\ 300,\ (\ :1),\ -0.989 - 0.145i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.00842987 + 0.115653i\)
\(L(\frac12)\) \(\approx\) \(0.00842987 + 0.115653i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.81 - 0.837i)T \)
3 \( 1 + (-1.32 + 2.69i)T \)
5 \( 1 \)
good7 \( 1 + (3.54 - 3.54i)T - 49iT^{2} \)
11 \( 1 + 16.8T + 121T^{2} \)
13 \( 1 + (-8.64 + 8.64i)T - 169iT^{2} \)
17 \( 1 + (9.72 - 9.72i)T - 289iT^{2} \)
19 \( 1 + 4.78T + 361T^{2} \)
23 \( 1 + (13.5 - 13.5i)T - 529iT^{2} \)
29 \( 1 + 14.8T + 841T^{2} \)
31 \( 1 - 14.0iT - 961T^{2} \)
37 \( 1 + (-10.1 - 10.1i)T + 1.36e3iT^{2} \)
41 \( 1 + 6.08iT - 1.68e3T^{2} \)
43 \( 1 + (57.2 + 57.2i)T + 1.84e3iT^{2} \)
47 \( 1 + (17.6 + 17.6i)T + 2.20e3iT^{2} \)
53 \( 1 + (16.2 + 16.2i)T + 2.80e3iT^{2} \)
59 \( 1 - 4.37iT - 3.48e3T^{2} \)
61 \( 1 - 8.52T + 3.72e3T^{2} \)
67 \( 1 + (53.9 - 53.9i)T - 4.48e3iT^{2} \)
71 \( 1 + 36.6T + 5.04e3T^{2} \)
73 \( 1 + (-12.6 + 12.6i)T - 5.32e3iT^{2} \)
79 \( 1 + 88.4T + 6.24e3T^{2} \)
83 \( 1 + (-63.7 + 63.7i)T - 6.88e3iT^{2} \)
89 \( 1 - 115.T + 7.92e3T^{2} \)
97 \( 1 + (-85.3 - 85.3i)T + 9.40e3iT^{2} \)
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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

−10.83878817273827973502853693553, −9.981503384486463608434450095583, −8.802865312075747323481823461474, −8.206799088214674633205794479050, −7.35040771813967941783208956161, −6.25142859634280573318188995804, −5.48127597614147652220472045473, −3.10155498467894770175284873638, −1.91705014830325994423440175893, −0.06515056102569562901271579350, 2.31628127913468617151662908780, 3.44125097168337626519440802154, 4.63237103275148692700058818153, 6.27761908828030604167455508281, 7.57654121893451326637344892352, 8.391495577280769409271571562393, 9.337910447574229809190537911660, 10.11515111015770812947107122227, 10.80067966119812030172175177385, 11.56238323703853881109861413603

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