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Properties

Label 2-300-4.3-c2-0-12
Degree $2$
Conductor $300$
Sign $0.866 - 0.5i$
Analytic cond. $8.17440$
Root an. cond. $2.85909$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 1.73i)2-s + 1.73i·3-s + (−1.99 − 3.46i)4-s + (2.99 + 1.73i)6-s + 10.3i·7-s − 7.99·8-s − 2.99·9-s + 10.3i·11-s + (5.99 − 3.46i)12-s + 18·13-s + (18 + 10.3i)14-s + (−8 + 13.8i)16-s + 10·17-s + (−2.99 + 5.19i)18-s + 13.8i·19-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + 0.577i·3-s + (−0.499 − 0.866i)4-s + (0.499 + 0.288i)6-s + 1.48i·7-s − 0.999·8-s − 0.333·9-s + 0.944i·11-s + (0.499 − 0.288i)12-s + 1.38·13-s + (1.28 + 0.742i)14-s + (−0.5 + 0.866i)16-s + 0.588·17-s + (−0.166 + 0.288i)18-s + 0.729i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.866 - 0.5i$
Analytic conductor: \(8.17440\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1),\ 0.866 - 0.5i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.76711 + 0.473496i\)
\(L(\frac12)\) \(\approx\) \(1.76711 + 0.473496i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + 1.73i)T \)
3 \( 1 - 1.73iT \)
5 \( 1 \)
good7 \( 1 - 10.3iT - 49T^{2} \)
11 \( 1 - 10.3iT - 121T^{2} \)
13 \( 1 - 18T + 169T^{2} \)
17 \( 1 - 10T + 289T^{2} \)
19 \( 1 - 13.8iT - 361T^{2} \)
23 \( 1 - 6.92iT - 529T^{2} \)
29 \( 1 + 36T + 841T^{2} \)
31 \( 1 + 6.92iT - 961T^{2} \)
37 \( 1 - 54T + 1.36e3T^{2} \)
41 \( 1 - 18T + 1.68e3T^{2} \)
43 \( 1 - 20.7iT - 1.84e3T^{2} \)
47 \( 1 - 2.20e3T^{2} \)
53 \( 1 + 26T + 2.80e3T^{2} \)
59 \( 1 + 31.1iT - 3.48e3T^{2} \)
61 \( 1 + 74T + 3.72e3T^{2} \)
67 \( 1 - 41.5iT - 4.48e3T^{2} \)
71 \( 1 + 103. iT - 5.04e3T^{2} \)
73 \( 1 - 36T + 5.32e3T^{2} \)
79 \( 1 - 90.0iT - 6.24e3T^{2} \)
83 \( 1 + 90.0iT - 6.88e3T^{2} \)
89 \( 1 + 18T + 7.92e3T^{2} \)
97 \( 1 + 72T + 9.40e3T^{2} \)
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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

−11.60234847954900137159105801100, −10.84868311889043295451243031892, −9.687317209334965565593538840097, −9.185844305479484554992814331839, −8.074054975502293242663106833871, −6.11657885591481747662234166419, −5.50416165200524974151139829966, −4.26911001390627901531722190622, −3.13068473769664397930115799277, −1.79625612887449332423523611722, 0.812249866827834641146803614656, 3.27355068860280451216097271988, 4.21367817798966557315072725753, 5.68917968192121087710075349027, 6.54122075787208288844131846417, 7.47795926774721656314499227414, 8.222119278958223583640863311540, 9.256410222527187871226993973719, 10.76020046130107952399319556028, 11.44641659161806970646949378119

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