Properties

Label 2-300-20.3-c1-0-7
Degree $2$
Conductor $300$
Sign $0.881 - 0.471i$
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  + (−0.581 + 1.28i)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 + (2.70 − 0.832i)8-s − 1.00i·9-s − 5.29i·11-s + (−1.99 − 0.125i)12-s + (3.74 + 3.74i)13-s + (−2.64 + 1.00i)14-s + (−0.5 + 3.96i)16-s + (1.28 + 0.581i)18-s + 5.29·19-s + 2.00·21-s + (6.82 + 3.07i)22-s + ⋯
L(s)  = 1  + (−0.411 + 0.911i)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 + (0.955 − 0.294i)8-s − 0.333i·9-s − 1.59i·11-s + (−0.576 − 0.0361i)12-s + (1.03 + 1.03i)13-s + (−0.707 + 0.267i)14-s + (−0.125 + 0.992i)16-s + (0.303 + 0.137i)18-s + 1.21·19-s + 0.436·21-s + (1.45 + 0.656i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.881 - 0.471i$
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: \((2,\ 300,\ (\ :1/2),\ 0.881 - 0.471i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21042 + 0.303219i\)
\(L(\frac12)\) \(\approx\) \(1.21042 + 0.303219i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.581 - 1.28i)T \)
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

−11.61304563941173601812870577347, −10.94855411512474847281763951691, −9.514105035182332663709989816895, −8.637984355792789576599075155623, −8.242969739085747641402215085963, −6.94607888999215966442474704990, −6.08216915231962127922487293835, −5.04005629367846542259151802536, −3.41890987961110218561738125590, −1.36346394600829757414452701235, 1.51836927040862045277958455948, 3.10535575240065756426573400207, 4.20997994512210031092797214015, 5.24033285927500502702530747982, 7.28687130677266467400413538053, 7.966382249863297798002835906320, 9.049090963698004712599244794623, 9.951962509561718602945257702883, 10.57661407920567291445529449357, 11.51508538374879549102542222095

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