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Properties

Label 2-300-20.7-c1-0-11
Degree $2$
Conductor $300$
Sign $0.923 - 0.382i$
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  + (1.22 + 0.707i)2-s + (−0.707 − 0.707i)3-s + (0.999 + 1.73i)4-s + (−0.366 − 1.36i)6-s + (3.15 − 3.15i)7-s + 2.82i·8-s + 1.00i·9-s + 2i·11-s + (0.517 − 1.93i)12-s + (1.60 − 1.60i)13-s + (6.09 − 1.63i)14-s + (−2.00 + 3.46i)16-s + (4.24 + 4.24i)17-s + (−0.707 + 1.22i)18-s − 3.19·19-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)2-s + (−0.408 − 0.408i)3-s + (0.499 + 0.866i)4-s + (−0.149 − 0.557i)6-s + (1.19 − 1.19i)7-s + 0.999i·8-s + 0.333i·9-s + 0.603i·11-s + (0.149 − 0.557i)12-s + (0.444 − 0.444i)13-s + (1.62 − 0.436i)14-s + (−0.500 + 0.866i)16-s + (1.02 + 1.02i)17-s + (−0.166 + 0.288i)18-s − 0.733·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.923 - 0.382i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.923 - 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.99178 + 0.395936i\)
\(L(\frac12)\) \(\approx\) \(1.99178 + 0.395936i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 - 0.707i)T \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 \)
good7 \( 1 + (-3.15 + 3.15i)T - 7iT^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 + (-1.60 + 1.60i)T - 13iT^{2} \)
17 \( 1 + (-4.24 - 4.24i)T + 17iT^{2} \)
19 \( 1 + 3.19T + 19T^{2} \)
23 \( 1 + (5.27 + 5.27i)T + 23iT^{2} \)
29 \( 1 - 0.535iT - 29T^{2} \)
31 \( 1 - 3.73iT - 31T^{2} \)
37 \( 1 + (7.72 + 7.72i)T + 37iT^{2} \)
41 \( 1 - 1.46T + 41T^{2} \)
43 \( 1 + (3.15 + 3.15i)T + 43iT^{2} \)
47 \( 1 + (0.656 - 0.656i)T - 47iT^{2} \)
53 \( 1 + (3.86 - 3.86i)T - 53iT^{2} \)
59 \( 1 + 11.4T + 59T^{2} \)
61 \( 1 - 3T + 61T^{2} \)
67 \( 1 + (3.91 - 3.91i)T - 67iT^{2} \)
71 \( 1 + 2.53iT - 71T^{2} \)
73 \( 1 + (2.82 - 2.82i)T - 73iT^{2} \)
79 \( 1 - 7.46T + 79T^{2} \)
83 \( 1 + (-9.14 - 9.14i)T + 83iT^{2} \)
89 \( 1 - 6.92iT - 89T^{2} \)
97 \( 1 + (6.88 + 6.88i)T + 97iT^{2} \)
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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

−12.14721725438499034749750919622, −10.83312100424646090270584759830, −10.48726391184732527156937809005, −8.390964352357426250324189037184, −7.78816741810436733350711274342, −6.87780449636749298470432089906, −5.81060862703871976984392050129, −4.70433408026730913379935274600, −3.80052576868758004327729803635, −1.78048616440873034354294552684, 1.77039408276890328389779016259, 3.29619556753170535485728675153, 4.66002522645508902310423901210, 5.47697921565076670252241917958, 6.25962119665459232926629672585, 7.86776973209365021565430803567, 9.038758930999932869463083503129, 10.03223086862074773755999617905, 11.16342007135298985103059804815, 11.69435245978603814277003251284

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