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Properties

Label 2-300-75.53-c1-0-9
Degree $2$
Conductor $300$
Sign $-0.395 + 0.918i$
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  + (0.670 − 1.59i)3-s + (−2.14 − 0.633i)5-s + (0.907 − 0.907i)7-s + (−2.10 − 2.14i)9-s + (−1.03 − 0.337i)11-s + (2.26 − 4.44i)13-s + (−2.45 + 2.99i)15-s + (−2.19 − 0.347i)17-s + (−2.39 − 3.30i)19-s + (−0.841 − 2.05i)21-s + (2.13 + 4.18i)23-s + (4.19 + 2.71i)25-s + (−4.82 + 1.91i)27-s + (−0.981 − 0.712i)29-s + (0.992 − 0.720i)31-s + ⋯
L(s)  = 1  + (0.387 − 0.921i)3-s + (−0.959 − 0.283i)5-s + (0.343 − 0.343i)7-s + (−0.700 − 0.713i)9-s + (−0.313 − 0.101i)11-s + (0.628 − 1.23i)13-s + (−0.632 + 0.774i)15-s + (−0.532 − 0.0843i)17-s + (−0.550 − 0.757i)19-s + (−0.183 − 0.449i)21-s + (0.445 + 0.873i)23-s + (0.839 + 0.543i)25-s + (−0.929 + 0.369i)27-s + (−0.182 − 0.132i)29-s + (0.178 − 0.129i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.395 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.395 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.395 + 0.918i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ -0.395 + 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.627901 - 0.954125i\)
\(L(\frac12)\) \(\approx\) \(0.627901 - 0.954125i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.670 + 1.59i)T \)
5 \( 1 + (2.14 + 0.633i)T \)
good7 \( 1 + (-0.907 + 0.907i)T - 7iT^{2} \)
11 \( 1 + (1.03 + 0.337i)T + (8.89 + 6.46i)T^{2} \)
13 \( 1 + (-2.26 + 4.44i)T + (-7.64 - 10.5i)T^{2} \)
17 \( 1 + (2.19 + 0.347i)T + (16.1 + 5.25i)T^{2} \)
19 \( 1 + (2.39 + 3.30i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + (-2.13 - 4.18i)T + (-13.5 + 18.6i)T^{2} \)
29 \( 1 + (0.981 + 0.712i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.992 + 0.720i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-6.47 - 3.29i)T + (21.7 + 29.9i)T^{2} \)
41 \( 1 + (-8.57 + 2.78i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + (-1.48 - 1.48i)T + 43iT^{2} \)
47 \( 1 + (0.0645 + 0.407i)T + (-44.6 + 14.5i)T^{2} \)
53 \( 1 + (-13.7 + 2.18i)T + (50.4 - 16.3i)T^{2} \)
59 \( 1 + (3.76 + 11.6i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-0.344 + 1.05i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (1.22 - 7.74i)T + (-63.7 - 20.7i)T^{2} \)
71 \( 1 + (-6.04 + 8.32i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-5.93 + 3.02i)T + (42.9 - 59.0i)T^{2} \)
79 \( 1 + (5.56 - 7.66i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (2.79 - 17.6i)T + (-78.9 - 25.6i)T^{2} \)
89 \( 1 + (4.10 - 12.6i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (15.3 - 2.42i)T + (92.2 - 29.9i)T^{2} \)
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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

−11.37428795069266384722615882190, −10.90313365063401927846384883430, −9.297048068641461252251031998792, −8.275033622491548216661747238732, −7.77036206803383206913955702738, −6.78586983440914806155844263655, −5.47069138792319682991193222848, −4.04412592277699080253289746808, −2.78002743060574680824462473704, −0.839810208650512861875347652706, 2.47242875031311437277959364132, 3.92909551969227069476521849657, 4.57677015593277589004513955456, 6.05703398733184144367436601599, 7.37869984169740006416681060142, 8.475147036059510918218860792005, 8.995438607689686202445808922613, 10.31213202734643800041825939607, 11.08110594677214376096772075777, 11.74274217658106455870281955100

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