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Properties

Label 2-300-60.47-c0-0-0
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

\(L(\frac{1}{2})\) \(\approx\) \(0.8733014658\)
\(L(\frac12)\) \(\approx\) \(0.8733014658\)
\(L(1)\) 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

−12.32326901821512692846905766774, −11.39757272760976552773564877687, −10.56423148582707130613583713992, −9.324723680575218994268603365427, −8.411918217483788657385467673099, −7.05624517925892448833598394965, −6.24012661431642841373249473491, −5.16295324392928414068576154021, −4.36061144814064941935865519036, −3.08218222097724472371337771954, 1.53297601872577559851715814642, 3.10217914592319341287474613947, 4.67850518427808979324199323128, 5.58502092796892536183335039452, 6.57414297771465698683497511585, 7.59317125952758812395903266448, 9.078018512143331325316058723892, 10.17221140688318340543047839763, 11.15039474022552947313958474407, 11.64699678709628673693167911071

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