Properties

Label 2-300-15.8-c1-0-1
Degree $2$
Conductor $300$
Sign $0.607 - 0.794i$
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  + (−1.61 + 0.618i)3-s + (1 − i)7-s + (2.23 − 2.00i)9-s + 4.47i·11-s + (3 + 3i)13-s + (2.23 + 2.23i)17-s + 2i·19-s + (−1 + 2.23i)21-s + (−2.23 + 2.23i)23-s + (−2.38 + 4.61i)27-s + 4.47·29-s + 4·31-s + (−2.76 − 7.23i)33-s + (3 − 3i)37-s + (−6.70 − 3i)39-s + ⋯
L(s)  = 1  + (−0.934 + 0.356i)3-s + (0.377 − 0.377i)7-s + (0.745 − 0.666i)9-s + 1.34i·11-s + (0.832 + 0.832i)13-s + (0.542 + 0.542i)17-s + 0.458i·19-s + (−0.218 + 0.487i)21-s + (−0.466 + 0.466i)23-s + (−0.458 + 0.888i)27-s + 0.830·29-s + 0.718·31-s + (−0.481 − 1.25i)33-s + (0.493 − 0.493i)37-s + (−1.07 − 0.480i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.607 - 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.607 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.607 - 0.794i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.607 - 0.794i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.910676 + 0.450308i\)
\(L(\frac12)\) \(\approx\) \(0.910676 + 0.450308i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.61 - 0.618i)T \)
5 \( 1 \)
good7 \( 1 + (-1 + i)T - 7iT^{2} \)
11 \( 1 - 4.47iT - 11T^{2} \)
13 \( 1 + (-3 - 3i)T + 13iT^{2} \)
17 \( 1 + (-2.23 - 2.23i)T + 17iT^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 + (2.23 - 2.23i)T - 23iT^{2} \)
29 \( 1 - 4.47T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (-3 + 3i)T - 37iT^{2} \)
41 \( 1 + 8.94iT - 41T^{2} \)
43 \( 1 + (-3 - 3i)T + 43iT^{2} \)
47 \( 1 + (6.70 + 6.70i)T + 47iT^{2} \)
53 \( 1 + (-2.23 + 2.23i)T - 53iT^{2} \)
59 \( 1 + 8.94T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + (-1 + i)T - 67iT^{2} \)
71 \( 1 + 4.47iT - 71T^{2} \)
73 \( 1 + (1 + i)T + 73iT^{2} \)
79 \( 1 - 6iT - 79T^{2} \)
83 \( 1 + (-6.70 + 6.70i)T - 83iT^{2} \)
89 \( 1 + 4.47T + 89T^{2} \)
97 \( 1 + (9 - 9i)T - 97iT^{2} \)
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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

−11.91643183053885871324136451364, −10.89425681136525979435902672482, −10.15980494105412588721091290465, −9.304052915595038920909501277771, −7.928900008981773018156893847854, −6.88018142651264645346305697017, −5.93317061011583505420907459422, −4.69823729292359967523597413842, −3.88826997094273363745296248787, −1.58007197925693039768896021020, 0.975701264454769177292084140345, 2.99439973037375114555512246422, 4.68610918289573643177172169771, 5.76113911921327072589666841386, 6.41670149233872565732193823297, 7.84159630611829227813236471232, 8.536673609276060105715686709295, 9.954181573427288442211915892205, 10.92963868845701674432784492390, 11.50456948901217277781713717769

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