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Properties

Label 2-300-15.14-c6-0-14
Degree $2$
Conductor $300$
Sign $-0.479 - 0.877i$
Analytic cond. $69.0162$
Root an. cond. $8.30760$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−15.3 + 22.1i)3-s + 437. i·7-s + (−254. − 683. i)9-s − 1.89e3i·11-s + 2.67e3i·13-s + 4.52e3·17-s + 9.50e3·19-s + (−9.69e3 − 6.73e3i)21-s − 549.·23-s + (1.90e4 + 4.86e3i)27-s + 4.33e4i·29-s + 2.95e4·31-s + (4.21e4 + 2.92e4i)33-s − 4.57e4i·37-s + (−5.92e4 − 4.11e4i)39-s + ⋯
L(s)  = 1  + (−0.570 + 0.821i)3-s + 1.27i·7-s + (−0.349 − 0.936i)9-s − 1.42i·11-s + 1.21i·13-s + 0.920·17-s + 1.38·19-s + (−1.04 − 0.726i)21-s − 0.0451·23-s + (0.968 + 0.247i)27-s + 1.77i·29-s + 0.993·31-s + (1.17 + 0.813i)33-s − 0.904i·37-s + (−0.998 − 0.693i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.479 - 0.877i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.479 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.479 - 0.877i$
Analytic conductor: \(69.0162\)
Root analytic conductor: \(8.30760\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :3),\ -0.479 - 0.877i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.606252017\)
\(L(\frac12)\) \(\approx\) \(1.606252017\)
\(L(4)\) 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 \( 1 + (15.3 - 22.1i)T \)
5 \( 1 \)
good7 \( 1 - 437. iT - 1.17e5T^{2} \)
11 \( 1 + 1.89e3iT - 1.77e6T^{2} \)
13 \( 1 - 2.67e3iT - 4.82e6T^{2} \)
17 \( 1 - 4.52e3T + 2.41e7T^{2} \)
19 \( 1 - 9.50e3T + 4.70e7T^{2} \)
23 \( 1 + 549.T + 1.48e8T^{2} \)
29 \( 1 - 4.33e4iT - 5.94e8T^{2} \)
31 \( 1 - 2.95e4T + 8.87e8T^{2} \)
37 \( 1 + 4.57e4iT - 2.56e9T^{2} \)
41 \( 1 + 1.14e5iT - 4.75e9T^{2} \)
43 \( 1 - 2.49e4iT - 6.32e9T^{2} \)
47 \( 1 + 1.82e4T + 1.07e10T^{2} \)
53 \( 1 - 1.74e5T + 2.21e10T^{2} \)
59 \( 1 - 6.78e4iT - 4.21e10T^{2} \)
61 \( 1 - 1.75e5T + 5.15e10T^{2} \)
67 \( 1 - 1.60e4iT - 9.04e10T^{2} \)
71 \( 1 - 2.22e5iT - 1.28e11T^{2} \)
73 \( 1 - 6.20e5iT - 1.51e11T^{2} \)
79 \( 1 + 5.25e5T + 2.43e11T^{2} \)
83 \( 1 + 7.90e5T + 3.26e11T^{2} \)
89 \( 1 - 1.12e6iT - 4.96e11T^{2} \)
97 \( 1 + 6.24e5iT - 8.32e11T^{2} \)
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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

−11.18423352324799434112747310312, −10.05022617507075239817165508297, −9.088993157861155933978107358269, −8.570048574120392868538054593209, −6.95146779329039711222360646316, −5.70805099450866277336958877057, −5.34117276261372593320416145268, −3.82536635321758813239717781004, −2.82400673741886644639938967663, −1.03844125842734020404306535586, 0.53112761624686922856362839832, 1.37605560986223792489279003956, 2.91457389941280279093859253911, 4.42011572765271726181666658053, 5.44008794932540067270323760674, 6.61723109445588713785211512512, 7.60432713690628586535895322876, 7.920044306731877350805463186558, 9.951761301462266477785726770757, 10.18398287829717777521023566695

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