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Properties

Label 2-300-4.3-c4-0-50
Degree $2$
Conductor $300$
Sign $0.0249 + 0.999i$
Analytic cond. $31.0109$
Root an. cond. $5.56875$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.99 + 0.0498i)2-s + 5.19i·3-s + (15.9 − 0.399i)4-s + (−0.259 − 20.7i)6-s + 35.2i·7-s + (−63.9 + 2.39i)8-s − 27·9-s + 3.95i·11-s + (2.07 + 83.1i)12-s − 41.0·13-s + (−1.76 − 141. i)14-s + (255. − 12.7i)16-s − 186.·17-s + (107. − 1.34i)18-s − 580. i·19-s + ⋯
L(s)  = 1  + (−0.999 + 0.0124i)2-s + 0.577i·3-s + (0.999 − 0.0249i)4-s + (−0.00720 − 0.577i)6-s + 0.720i·7-s + (−0.999 + 0.0374i)8-s − 0.333·9-s + 0.0326i·11-s + (0.0144 + 0.577i)12-s − 0.242·13-s + (−0.00898 − 0.720i)14-s + (0.998 − 0.0498i)16-s − 0.646·17-s + (0.333 − 0.00415i)18-s − 1.60i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0249 + 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0249 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.0249 + 0.999i$
Analytic conductor: \(31.0109\)
Root analytic conductor: \(5.56875\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :2),\ 0.0249 + 0.999i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.3983722761\)
\(L(\frac12)\) \(\approx\) \(0.3983722761\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.99 - 0.0498i)T \)
3 \( 1 - 5.19iT \)
5 \( 1 \)
good7 \( 1 - 35.2iT - 2.40e3T^{2} \)
11 \( 1 - 3.95iT - 1.46e4T^{2} \)
13 \( 1 + 41.0T + 2.85e4T^{2} \)
17 \( 1 + 186.T + 8.35e4T^{2} \)
19 \( 1 + 580. iT - 1.30e5T^{2} \)
23 \( 1 - 472. iT - 2.79e5T^{2} \)
29 \( 1 + 979.T + 7.07e5T^{2} \)
31 \( 1 - 33.2iT - 9.23e5T^{2} \)
37 \( 1 - 623.T + 1.87e6T^{2} \)
41 \( 1 - 154.T + 2.82e6T^{2} \)
43 \( 1 - 2.67e3iT - 3.41e6T^{2} \)
47 \( 1 + 2.92e3iT - 4.87e6T^{2} \)
53 \( 1 + 3.78e3T + 7.89e6T^{2} \)
59 \( 1 + 3.48e3iT - 1.21e7T^{2} \)
61 \( 1 + 488.T + 1.38e7T^{2} \)
67 \( 1 + 7.49e3iT - 2.01e7T^{2} \)
71 \( 1 + 4.62e3iT - 2.54e7T^{2} \)
73 \( 1 - 6.95e3T + 2.83e7T^{2} \)
79 \( 1 + 6.75e3iT - 3.89e7T^{2} \)
83 \( 1 + 4.86e3iT - 4.74e7T^{2} \)
89 \( 1 - 513.T + 6.27e7T^{2} \)
97 \( 1 + 1.21e4T + 8.85e7T^{2} \)
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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

−10.95144525654144526459209629271, −9.570874821663161176334901607921, −9.243672640734231678870690514338, −8.212861114974342149148233635254, −7.12057266976771983991656424040, −6.02640546370453457796252922142, −4.87563540933838125445710894625, −3.19483432724174319001472391238, −2.03194809874762860367236857431, −0.18066829528717244269768664600, 1.15660790643076058929157854130, 2.38310436582820856417236308754, 3.90625516153246976235812961988, 5.72804786503935452569442568753, 6.74025799053828848674777689438, 7.58176179706430560974351957868, 8.361331538175573966885209521024, 9.431468130246479332858163032847, 10.39739051095758142608240976591, 11.12072994429452298605324775588

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