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Properties

Label 2-300-20.3-c1-0-14
Degree $2$
Conductor $300$
Sign $0.935 + 0.353i$
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  + (1.41 + 0.0912i)2-s + (0.707 − 0.707i)3-s + (1.98 + 0.257i)4-s + (1.06 − 0.933i)6-s + (−1.86 − 1.86i)7-s + (2.77 + 0.544i)8-s − 1.00i·9-s − 0.728i·11-s + (1.58 − 1.22i)12-s + (3.12 + 3.12i)13-s + (−2.46 − 2.80i)14-s + (3.86 + 1.02i)16-s + (−1.12 + 1.12i)17-s + (0.0912 − 1.41i)18-s − 3.73·19-s + ⋯
L(s)  = 1  + (0.997 + 0.0645i)2-s + (0.408 − 0.408i)3-s + (0.991 + 0.128i)4-s + (0.433 − 0.381i)6-s + (−0.705 − 0.705i)7-s + (0.981 + 0.192i)8-s − 0.333i·9-s − 0.219i·11-s + (0.457 − 0.352i)12-s + (0.866 + 0.866i)13-s + (−0.658 − 0.749i)14-s + (0.966 + 0.255i)16-s + (−0.272 + 0.272i)17-s + (0.0215 − 0.332i)18-s − 0.856·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.353i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.935 + 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.935 + 0.353i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.935 + 0.353i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.45651 - 0.448237i\)
\(L(\frac12)\) \(\approx\) \(2.45651 - 0.448237i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 - 0.0912i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 \)
good7 \( 1 + (1.86 + 1.86i)T + 7iT^{2} \)
11 \( 1 + 0.728iT - 11T^{2} \)
13 \( 1 + (-3.12 - 3.12i)T + 13iT^{2} \)
17 \( 1 + (1.12 - 1.12i)T - 17iT^{2} \)
19 \( 1 + 3.73T + 19T^{2} \)
23 \( 1 + (5.83 - 5.83i)T - 23iT^{2} \)
29 \( 1 - 2.64iT - 29T^{2} \)
31 \( 1 + 6.01iT - 31T^{2} \)
37 \( 1 + (3.12 - 3.12i)T - 37iT^{2} \)
41 \( 1 + 4.24T + 41T^{2} \)
43 \( 1 + (-5.10 + 5.10i)T - 43iT^{2} \)
47 \( 1 + (-2.09 - 2.09i)T + 47iT^{2} \)
53 \( 1 + (0.484 + 0.484i)T + 53iT^{2} \)
59 \( 1 + 4.92T + 59T^{2} \)
61 \( 1 - 2.31T + 61T^{2} \)
67 \( 1 + (5.10 + 5.10i)T + 67iT^{2} \)
71 \( 1 - 13.1iT - 71T^{2} \)
73 \( 1 + (3.96 + 3.96i)T + 73iT^{2} \)
79 \( 1 - 7.11T + 79T^{2} \)
83 \( 1 + (-3.55 + 3.55i)T - 83iT^{2} \)
89 \( 1 - 1.03iT - 89T^{2} \)
97 \( 1 + (-12.5 + 12.5i)T - 97iT^{2} \)
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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

−11.86690265560310636061354602423, −10.97898454053806892071646820662, −9.973600936185577405934416491309, −8.692113242809416841517169330880, −7.56592771105325254996895923216, −6.63922695400522886198707801321, −5.90057937465302851409636266789, −4.20930489190661362563649851268, −3.46971142264120897335295565981, −1.86933103756159213637693505089, 2.33232371828655254728883350738, 3.41693746278394688542196685161, 4.53491627440300939593984478703, 5.79250926073661343182413766094, 6.55755673690600485416007222732, 7.966524003898824239332725357981, 8.944336224873259683050560261803, 10.20058033812286226404152914171, 10.81645850640574664819495888791, 12.09251253779282533122077485385

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