Properties

Label 2-300-20.3-c1-0-10
Degree $2$
Conductor $300$
Sign $0.699 + 0.715i$
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  + (−0.394 + 1.35i)2-s + (−0.707 + 0.707i)3-s + (−1.68 − 1.07i)4-s + (−0.681 − 1.23i)6-s + (−2.47 − 2.47i)7-s + (2.11 − 1.87i)8-s − 1.00i·9-s − 3.02i·11-s + (1.95 − 0.437i)12-s + (−0.363 − 0.363i)13-s + (4.34 − 2.38i)14-s + (1.70 + 3.61i)16-s + (2.36 − 2.36i)17-s + (1.35 + 0.394i)18-s − 4.95·19-s + ⋯
L(s)  = 1  + (−0.278 + 0.960i)2-s + (−0.408 + 0.408i)3-s + (−0.844 − 0.535i)4-s + (−0.278 − 0.505i)6-s + (−0.936 − 0.936i)7-s + (0.749 − 0.661i)8-s − 0.333i·9-s − 0.913i·11-s + (0.563 − 0.126i)12-s + (−0.100 − 0.100i)13-s + (1.16 − 0.638i)14-s + (0.426 + 0.904i)16-s + (0.573 − 0.573i)17-s + (0.320 + 0.0929i)18-s − 1.13·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 + 0.715i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.699 + 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.699 + 0.715i$
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: \((2,\ 300,\ (\ :1/2),\ 0.699 + 0.715i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.486395 - 0.204707i\)
\(L(\frac12)\) \(\approx\) \(0.486395 - 0.204707i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.394 - 1.35i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 \)
good7 \( 1 + (2.47 + 2.47i)T + 7iT^{2} \)
11 \( 1 + 3.02iT - 11T^{2} \)
13 \( 1 + (0.363 + 0.363i)T + 13iT^{2} \)
17 \( 1 + (-2.36 + 2.36i)T - 17iT^{2} \)
19 \( 1 + 4.95T + 19T^{2} \)
23 \( 1 + (-0.900 + 0.900i)T - 23iT^{2} \)
29 \( 1 + 3.50iT - 29T^{2} \)
31 \( 1 + 3.85iT - 31T^{2} \)
37 \( 1 + (-0.363 + 0.363i)T - 37iT^{2} \)
41 \( 1 - 2.72T + 41T^{2} \)
43 \( 1 + (3.92 - 3.92i)T - 43iT^{2} \)
47 \( 1 + (5.85 + 5.85i)T + 47iT^{2} \)
53 \( 1 + (3.14 + 3.14i)T + 53iT^{2} \)
59 \( 1 - 8.68T + 59T^{2} \)
61 \( 1 + 15.2T + 61T^{2} \)
67 \( 1 + (-3.92 - 3.92i)T + 67iT^{2} \)
71 \( 1 - 4.25iT - 71T^{2} \)
73 \( 1 + (9.28 + 9.28i)T + 73iT^{2} \)
79 \( 1 - 0.399T + 79T^{2} \)
83 \( 1 + (-0.199 + 0.199i)T - 83iT^{2} \)
89 \( 1 + 4.28iT - 89T^{2} \)
97 \( 1 + (6.73 - 6.73i)T - 97iT^{2} \)
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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

−11.40366869791224029362024587431, −10.37163566039483490382815148348, −9.796878668348751713613696582525, −8.756770191963503834993846243885, −7.66733212748747165952650720973, −6.61134774835641381139430384153, −5.89903266044339125563358531736, −4.61446306108342766593001648957, −3.50648444494165044487245167749, −0.44836627200866027856961988057, 1.84935072363731228825814436478, 3.13535207034008008661760829833, 4.61245206600731883836678087309, 5.86506378648367610136617221342, 7.03433407280178783032891060915, 8.284171906959321661976807132996, 9.222410692287019126704055095368, 10.06423987747900787740259552467, 10.93269140889144926844964891062, 12.07857282633881876155910648027

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