Properties

Label 2-300-4.3-c2-0-34
Degree $2$
Conductor $300$
Sign $0.168 + 0.985i$
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  + (1.99 − 0.169i)2-s − 1.73i·3-s + (3.94 − 0.675i)4-s + (−0.293 − 3.45i)6-s − 12.3i·7-s + (7.74 − 2.01i)8-s − 2.99·9-s + 11.0i·11-s + (−1.16 − 6.82i)12-s − 2.82·13-s + (−2.10 − 24.7i)14-s + (15.0 − 5.32i)16-s − 6.52·17-s + (−5.97 + 0.508i)18-s − 27.9i·19-s + ⋯
L(s)  = 1  + (0.996 − 0.0847i)2-s − 0.577i·3-s + (0.985 − 0.168i)4-s + (−0.0489 − 0.575i)6-s − 1.77i·7-s + (0.967 − 0.251i)8-s − 0.333·9-s + 1.00i·11-s + (−0.0974 − 0.569i)12-s − 0.216·13-s + (−0.150 − 1.76i)14-s + (0.942 − 0.332i)16-s − 0.383·17-s + (−0.332 + 0.0282i)18-s − 1.47i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.168 + 0.985i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.168 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.168 + 0.985i$
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: \((2,\ 300,\ (\ :1),\ 0.168 + 0.985i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.27321 - 1.91693i\)
\(L(\frac12)\) \(\approx\) \(2.27321 - 1.91693i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.99 + 0.169i)T \)
3 \( 1 + 1.73iT \)
5 \( 1 \)
good7 \( 1 + 12.3iT - 49T^{2} \)
11 \( 1 - 11.0iT - 121T^{2} \)
13 \( 1 + 2.82T + 169T^{2} \)
17 \( 1 + 6.52T + 289T^{2} \)
19 \( 1 + 27.9iT - 361T^{2} \)
23 \( 1 - 7.90iT - 529T^{2} \)
29 \( 1 - 50.7T + 841T^{2} \)
31 \( 1 - 36.3iT - 961T^{2} \)
37 \( 1 - 18.9T + 1.36e3T^{2} \)
41 \( 1 - 5.30T + 1.68e3T^{2} \)
43 \( 1 - 45.5iT - 1.84e3T^{2} \)
47 \( 1 + 11.7iT - 2.20e3T^{2} \)
53 \( 1 + 41.1T + 2.80e3T^{2} \)
59 \( 1 - 10.7iT - 3.48e3T^{2} \)
61 \( 1 - 56.1T + 3.72e3T^{2} \)
67 \( 1 - 16.1iT - 4.48e3T^{2} \)
71 \( 1 - 66.1iT - 5.04e3T^{2} \)
73 \( 1 + 15.6T + 5.32e3T^{2} \)
79 \( 1 - 123. iT - 6.24e3T^{2} \)
83 \( 1 - 99.6iT - 6.88e3T^{2} \)
89 \( 1 - 101.T + 7.92e3T^{2} \)
97 \( 1 + 127.T + 9.40e3T^{2} \)
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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

−11.40267776648102667285962901210, −10.64394931566110909266269473657, −9.743158764543393729036676935371, −8.026127869561781486906620284565, −6.95543456679686346618312360136, −6.75442940324712039315169348314, −4.96220299339527526628964254346, −4.20537693342380536323840486745, −2.76052223612923344602076858706, −1.15391622641662921526962140105, 2.29731079493737612065341272348, 3.34667578868649304567467528669, 4.69244066526921682013663499190, 5.76075754567050115038566140064, 6.25368428060902073877598584942, 8.009581653218328742810014986709, 8.771943706499456742695856333735, 9.992725978226262075728399974994, 11.06338011746953277783516918635, 11.91638480428719063258808068363

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