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Properties

Label 2-300-5.4-c3-0-4
Degree $2$
Conductor $300$
Sign $0.894 - 0.447i$
Analytic cond. $17.7005$
Root an. cond. $4.20720$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3i·3-s − 8i·7-s − 9·9-s + 36·11-s − 10i·13-s − 18i·17-s + 100·19-s + 24·21-s + 72i·23-s − 27i·27-s + 234·29-s − 16·31-s + 108i·33-s + 226i·37-s + 30·39-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.431i·7-s − 0.333·9-s + 0.986·11-s − 0.213i·13-s − 0.256i·17-s + 1.20·19-s + 0.249·21-s + 0.652i·23-s − 0.192i·27-s + 1.49·29-s − 0.0926·31-s + 0.569i·33-s + 1.00i·37-s + 0.123·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(17.7005\)
Root analytic conductor: \(4.20720\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :3/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.953788665\)
\(L(\frac12)\) \(\approx\) \(1.953788665\)
\(L(\frac{5}{2})\) 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 \( 1 - 3iT \)
5 \( 1 \)
good7 \( 1 + 8iT - 343T^{2} \)
11 \( 1 - 36T + 1.33e3T^{2} \)
13 \( 1 + 10iT - 2.19e3T^{2} \)
17 \( 1 + 18iT - 4.91e3T^{2} \)
19 \( 1 - 100T + 6.85e3T^{2} \)
23 \( 1 - 72iT - 1.21e4T^{2} \)
29 \( 1 - 234T + 2.43e4T^{2} \)
31 \( 1 + 16T + 2.97e4T^{2} \)
37 \( 1 - 226iT - 5.06e4T^{2} \)
41 \( 1 - 90T + 6.89e4T^{2} \)
43 \( 1 - 452iT - 7.95e4T^{2} \)
47 \( 1 + 432iT - 1.03e5T^{2} \)
53 \( 1 - 414iT - 1.48e5T^{2} \)
59 \( 1 - 684T + 2.05e5T^{2} \)
61 \( 1 - 422T + 2.26e5T^{2} \)
67 \( 1 + 332iT - 3.00e5T^{2} \)
71 \( 1 + 360T + 3.57e5T^{2} \)
73 \( 1 - 26iT - 3.89e5T^{2} \)
79 \( 1 + 512T + 4.93e5T^{2} \)
83 \( 1 + 1.18e3iT - 5.71e5T^{2} \)
89 \( 1 - 630T + 7.04e5T^{2} \)
97 \( 1 - 1.05e3iT - 9.12e5T^{2} \)
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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

−11.46034422609043977428802931929, −10.30123159622359384634317176678, −9.614038276757062800617854573237, −8.658083341052038512888120794330, −7.51242743499894942596596104057, −6.45189725022693994664513603490, −5.23191986077072422926754362984, −4.13521002902503709649179080873, −3.03226289994440797880429787985, −1.07799533821304888474919268607, 0.998695260928574137132394058307, 2.45927099866242398519438737243, 3.89777376088361802418684514920, 5.33101338037552843692236606174, 6.40216532635018903090561939321, 7.25918426715692784224466813855, 8.440588393978144737884839762398, 9.201726319708660455400379578182, 10.30149725236378895947524459441, 11.51536405683585010970300149040

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