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Properties

Label 2-300-60.47-c0-0-1
Degree $2$
Conductor $300$
Sign $0.229 + 0.973i$
Analytic cond. $0.149719$
Root an. cond. $0.386936$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)2-s + (0.707 − 0.707i)3-s + 1.00i·4-s − 1.00·6-s + (0.707 − 0.707i)8-s − 1.00i·9-s + (0.707 + 0.707i)12-s − 1.00·16-s + (−0.707 + 0.707i)18-s + (−1.41 + 1.41i)23-s − 1.00i·24-s + (−0.707 − 0.707i)27-s + (0.707 + 0.707i)32-s + 1.00·36-s + 2.00·46-s + (1.41 + 1.41i)47-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + (0.707 − 0.707i)3-s + 1.00i·4-s − 1.00·6-s + (0.707 − 0.707i)8-s − 1.00i·9-s + (0.707 + 0.707i)12-s − 1.00·16-s + (−0.707 + 0.707i)18-s + (−1.41 + 1.41i)23-s − 1.00i·24-s + (−0.707 − 0.707i)27-s + (0.707 + 0.707i)32-s + 1.00·36-s + 2.00·46-s + (1.41 + 1.41i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.229 + 0.973i$
Analytic conductor: \(0.149719\)
Root analytic conductor: \(0.386936\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :0),\ 0.229 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6786550477\)
\(L(\frac12)\) \(\approx\) \(0.6786550477\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 \)
good7 \( 1 - iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 2T + T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + iT^{2} \)
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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

−11.89337111932262939459546795726, −10.82697880088106251299296589853, −9.698795883592432429217393018388, −9.043328147972449035216947636172, −7.959230405478041662023382150220, −7.39702987660357997151345068660, −6.08961050487577342783472551019, −4.11189807717108522972642807070, −2.95202118631667417301243876022, −1.64326021482888890759658743212, 2.27254191941686987914538653412, 4.05799487665792761554002855629, 5.20514727985965618683528124107, 6.41489599425655357514547952408, 7.63682054159941470575817119598, 8.432854045302240909782223598137, 9.206753516558630516183198582758, 10.17969104465293479269883107173, 10.71344051272530761808112858427, 12.02717060312634771820883431067

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