Properties

Label 2-300-4.3-c2-0-19
Degree $2$
Conductor $300$
Sign $0.302 - 0.953i$
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  + (1.97 + 0.305i)2-s + 1.73i·3-s + (3.81 + 1.20i)4-s + (−0.529 + 3.42i)6-s − 0.329i·7-s + (7.16 + 3.55i)8-s − 2.99·9-s + 20.4i·11-s + (−2.09 + 6.60i)12-s + 0.416·13-s + (0.100 − 0.652i)14-s + (13.0 + 9.21i)16-s + 18.5·17-s + (−5.92 − 0.917i)18-s − 12.4i·19-s + ⋯
L(s)  = 1  + (0.988 + 0.152i)2-s + 0.577i·3-s + (0.953 + 0.302i)4-s + (−0.0882 + 0.570i)6-s − 0.0471i·7-s + (0.895 + 0.444i)8-s − 0.333·9-s + 1.86i·11-s + (−0.174 + 0.550i)12-s + 0.0320·13-s + (0.00720 − 0.0465i)14-s + (0.817 + 0.575i)16-s + 1.09·17-s + (−0.329 − 0.0509i)18-s − 0.655i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.302 - 0.953i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.302 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.302 - 0.953i$
Analytic conductor: \(8.17440\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1),\ 0.302 - 0.953i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.43846 + 1.78525i\)
\(L(\frac12)\) \(\approx\) \(2.43846 + 1.78525i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.97 - 0.305i)T \)
3 \( 1 - 1.73iT \)
5 \( 1 \)
good7 \( 1 + 0.329iT - 49T^{2} \)
11 \( 1 - 20.4iT - 121T^{2} \)
13 \( 1 - 0.416T + 169T^{2} \)
17 \( 1 - 18.5T + 289T^{2} \)
19 \( 1 + 12.4iT - 361T^{2} \)
23 \( 1 + 23.2iT - 529T^{2} \)
29 \( 1 + 23.9T + 841T^{2} \)
31 \( 1 - 42.0iT - 961T^{2} \)
37 \( 1 + 50.9T + 1.36e3T^{2} \)
41 \( 1 - 46.7T + 1.68e3T^{2} \)
43 \( 1 + 55.5iT - 1.84e3T^{2} \)
47 \( 1 + 81.7iT - 2.20e3T^{2} \)
53 \( 1 - 29.9T + 2.80e3T^{2} \)
59 \( 1 + 24.3iT - 3.48e3T^{2} \)
61 \( 1 + 74.8T + 3.72e3T^{2} \)
67 \( 1 + 72.8iT - 4.48e3T^{2} \)
71 \( 1 + 39.2iT - 5.04e3T^{2} \)
73 \( 1 - 46.5T + 5.32e3T^{2} \)
79 \( 1 + 101. iT - 6.24e3T^{2} \)
83 \( 1 + 5.88iT - 6.88e3T^{2} \)
89 \( 1 + 61.0T + 7.92e3T^{2} \)
97 \( 1 + 95.5T + 9.40e3T^{2} \)
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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

−12.13353251888163776377843595404, −10.72180981348719882457329017271, −10.13593117150966288902783328822, −8.906240313041634457908524033358, −7.53034872006517589021444196844, −6.79791410030733304260401106921, −5.39536487947684389457746607559, −4.64003489282041583741772376179, −3.54435717562605980028906497702, −2.10800363915412218645759674755, 1.21455868163012998963015860871, 2.89516647211867193655260175549, 3.86896405859782586109365412912, 5.66003183566118909486910837749, 5.93118075237723909975771740424, 7.37386621669315062710401180000, 8.168738307287176169180833362184, 9.559122778255918412958009739497, 10.84702426228794631065818524033, 11.46065545718790226868376739943

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