Properties

Label 2-300-12.11-c1-0-4
Degree $2$
Conductor $300$
Sign $-0.408 - 0.912i$
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  − 1.41·2-s + (0.707 + 1.58i)3-s + 2.00·4-s + (−1.00 − 2.23i)6-s + 3.16i·7-s − 2.82·8-s + (−2.00 + 2.23i)9-s + (1.41 + 3.16i)12-s − 4.47i·14-s + 4.00·16-s + (2.82 − 3.16i)18-s + (−5.00 + 2.23i)21-s − 1.41·23-s + (−2.00 − 4.47i)24-s + (−4.94 − 1.58i)27-s + 6.32i·28-s + ⋯
L(s)  = 1  − 1.00·2-s + (0.408 + 0.912i)3-s + 1.00·4-s + (−0.408 − 0.912i)6-s + 1.19i·7-s − 1.00·8-s + (−0.666 + 0.745i)9-s + (0.408 + 0.912i)12-s − 1.19i·14-s + 1.00·16-s + (0.666 − 0.745i)18-s + (−1.09 + 0.487i)21-s − 0.294·23-s + (−0.408 − 0.912i)24-s + (−0.952 − 0.304i)27-s + 1.19i·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.408 - 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.408 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.408 - 0.912i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ -0.408 - 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.464894 + 0.717173i\)
\(L(\frac12)\) \(\approx\) \(0.464894 + 0.717173i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
3 \( 1 + (-0.707 - 1.58i)T \)
5 \( 1 \)
good7 \( 1 - 3.16iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 1.41T + 23T^{2} \)
29 \( 1 - 8.94iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 4.47iT - 41T^{2} \)
43 \( 1 + 3.16iT - 43T^{2} \)
47 \( 1 - 9.89T + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + 15.8iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 15.5T + 83T^{2} \)
89 \( 1 + 17.8iT - 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−11.75675778843100714566471203762, −10.86487665919189777262031032586, −10.02909412493151551553869249102, −9.060520753114980386466674421723, −8.651298742971363718079070587365, −7.54662643931072457699598369477, −6.13738944215972820559193889449, −5.11244926398844520313607032892, −3.36592430985495431315241637102, −2.20948265216510993730510528330, 0.826896808199285001952706185328, 2.34879477507242529031403756799, 3.83745552114054671214763759206, 5.92609784288548712531003643413, 6.94745777030392306371897200553, 7.62346796558641939159658080491, 8.430185419055942607208072514309, 9.507040536944128564755309985589, 10.40023164374517050993345475509, 11.37708181572973165851678757763

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