Properties

Label 2-300-20.3-c1-0-15
Degree $2$
Conductor $300$
Sign $0.577 + 0.816i$
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  + (1.28 − 0.581i)2-s + (0.707 − 0.707i)3-s + (1.32 − 1.50i)4-s + (0.500 − 1.32i)6-s + (1.41 + 1.41i)7-s + (0.832 − 2.70i)8-s − 1.00i·9-s + 5.29i·11-s + (−0.125 − 1.99i)12-s + (−3.74 − 3.74i)13-s + (2.64 + 1.00i)14-s + (−0.5 − 3.96i)16-s + (−0.581 − 1.28i)18-s − 5.29·19-s + 2.00·21-s + (3.07 + 6.82i)22-s + ⋯
L(s)  = 1  + (0.911 − 0.411i)2-s + (0.408 − 0.408i)3-s + (0.661 − 0.750i)4-s + (0.204 − 0.540i)6-s + (0.534 + 0.534i)7-s + (0.294 − 0.955i)8-s − 0.333i·9-s + 1.59i·11-s + (−0.0361 − 0.576i)12-s + (−1.03 − 1.03i)13-s + (0.707 + 0.267i)14-s + (−0.125 − 0.992i)16-s + (−0.137 − 0.303i)18-s − 1.21·19-s + 0.436·21-s + (0.656 + 1.45i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.577 + 0.816i$
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: \((2,\ 300,\ (\ :1/2),\ 0.577 + 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.15816 - 1.11611i\)
\(L(\frac12)\) \(\approx\) \(2.15816 - 1.11611i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.28 + 0.581i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 \)
good7 \( 1 + (-1.41 - 1.41i)T + 7iT^{2} \)
11 \( 1 - 5.29iT - 11T^{2} \)
13 \( 1 + (3.74 + 3.74i)T + 13iT^{2} \)
17 \( 1 - 17iT^{2} \)
19 \( 1 + 5.29T + 19T^{2} \)
23 \( 1 + (-2.82 + 2.82i)T - 23iT^{2} \)
29 \( 1 - 8iT - 29T^{2} \)
31 \( 1 - 5.29iT - 31T^{2} \)
37 \( 1 + (-3.74 + 3.74i)T - 37iT^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + (5.65 - 5.65i)T - 43iT^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 + (-7.48 - 7.48i)T + 53iT^{2} \)
59 \( 1 - 5.29T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 + (8.48 + 8.48i)T + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (7.48 + 7.48i)T + 73iT^{2} \)
79 \( 1 - 5.29T + 79T^{2} \)
83 \( 1 + (8.48 - 8.48i)T - 83iT^{2} \)
89 \( 1 + 6iT - 89T^{2} \)
97 \( 1 + (-7.48 + 7.48i)T - 97iT^{2} \)
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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

−12.00261544903505866110306947231, −10.72475686663289832532216447691, −9.977648115084552895613475820998, −8.774297464783785001142892640421, −7.50368167143674747354241170255, −6.71626927742560605836573337902, −5.28811650005377487617963425505, −4.51072559065223491151326086084, −2.87460730411740271355271987780, −1.86993810611148617767534413032, 2.38318431600931970056410301824, 3.78882890641502550285714123177, 4.62188967479394308618022846899, 5.81786391316096101299750726651, 6.94499975048256863381131109278, 7.998479774653973052518595643797, 8.783267865266223166708670330192, 10.13002690183170937462681156926, 11.28909031293119663743300521639, 11.70647034716235311966839185152

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