Properties

Label 2-300-60.59-c1-0-1
Degree $2$
Conductor $300$
Sign $0.345 - 0.938i$
Analytic cond. $2.39551$
Root an. cond. $1.54774$
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  + (−0.780 − 1.17i)2-s + (−0.848 − 1.51i)3-s + (−0.780 + 1.84i)4-s + (−1.11 + 2.17i)6-s − 3.02·7-s + (2.78 − 0.516i)8-s + (−1.56 + 2.56i)9-s − 1.32·11-s + (3.44 − 0.382i)12-s + 5.12i·13-s + (2.35 + 3.56i)14-s + (−2.78 − 2.87i)16-s − 2·17-s + (4.23 − 0.158i)18-s + 1.32i·19-s + ⋯
L(s)  = 1  + (−0.552 − 0.833i)2-s + (−0.489 − 0.871i)3-s + (−0.390 + 0.920i)4-s + (−0.456 + 0.889i)6-s − 1.14·7-s + (0.983 − 0.182i)8-s + (−0.520 + 0.853i)9-s − 0.399·11-s + (0.993 − 0.110i)12-s + 1.42i·13-s + (0.630 + 0.951i)14-s + (−0.695 − 0.718i)16-s − 0.485·17-s + (0.999 − 0.0374i)18-s + 0.303i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.345 - 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.345 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.345 - 0.938i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.345 - 0.938i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.161618 + 0.112689i\)
\(L(\frac12)\) \(\approx\) \(0.161618 + 0.112689i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.780 + 1.17i)T \)
3 \( 1 + (0.848 + 1.51i)T \)
5 \( 1 \)
good7 \( 1 + 3.02T + 7T^{2} \)
11 \( 1 + 1.32T + 11T^{2} \)
13 \( 1 - 5.12iT - 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 1.32iT - 19T^{2} \)
23 \( 1 - 0.371iT - 23T^{2} \)
29 \( 1 + 3.12iT - 29T^{2} \)
31 \( 1 - 4.71iT - 31T^{2} \)
37 \( 1 - 5.12iT - 37T^{2} \)
41 \( 1 + 1.12iT - 41T^{2} \)
43 \( 1 + 7.73T + 43T^{2} \)
47 \( 1 + 3.02iT - 47T^{2} \)
53 \( 1 + 12.2T + 53T^{2} \)
59 \( 1 + 14.1T + 59T^{2} \)
61 \( 1 - 3.12T + 61T^{2} \)
67 \( 1 + 4.34T + 67T^{2} \)
71 \( 1 + 3.39T + 71T^{2} \)
73 \( 1 + 8.24iT - 73T^{2} \)
79 \( 1 - 8.10iT - 79T^{2} \)
83 \( 1 - 15.1iT - 83T^{2} \)
89 \( 1 + 10.2iT - 89T^{2} \)
97 \( 1 + 6iT - 97T^{2} \)
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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

−11.90959532223163703476003397541, −11.11746012449375687520430891651, −10.11989374102312885484061589975, −9.216862767422867000465354859210, −8.207827177191409285389863105112, −7.06758886856825459342118321715, −6.32121715901158913471381154930, −4.67504970126721101903202572465, −3.14658849291479003509065990565, −1.79545870087808403391718245635, 0.17793161062741461146819179589, 3.19765502750783427585763843098, 4.70112310348902705814701297858, 5.73518824259380817496234678422, 6.48401813699305624377246618866, 7.71595610271337275482079222889, 8.871701516862702136062592573465, 9.686818818020277284931698524774, 10.38264813344574388538024534399, 11.12937770452053077239195847994

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