Properties

Label 2-300-20.19-c2-0-7
Degree $2$
Conductor $300$
Sign $-0.344 - 0.938i$
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  + (−1.49 + 1.33i)2-s + 1.73·3-s + (0.446 − 3.97i)4-s + (−2.58 + 2.30i)6-s − 6.56·7-s + (4.63 + 6.52i)8-s + 2.99·9-s − 2.26i·11-s + (0.773 − 6.88i)12-s + 14.8i·13-s + (9.79 − 8.75i)14-s + (−15.6 − 3.55i)16-s + 26.8i·17-s + (−4.47 + 3.99i)18-s + 10.8i·19-s + ⋯
L(s)  = 1  + (−0.745 + 0.666i)2-s + 0.577·3-s + (0.111 − 0.993i)4-s + (−0.430 + 0.384i)6-s − 0.938·7-s + (0.579 + 0.815i)8-s + 0.333·9-s − 0.206i·11-s + (0.0644 − 0.573i)12-s + 1.14i·13-s + (0.699 − 0.625i)14-s + (−0.975 − 0.221i)16-s + 1.57i·17-s + (−0.248 + 0.222i)18-s + 0.572i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.344 - 0.938i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.344 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.344 - 0.938i$
Analytic conductor: \(8.17440\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1),\ -0.344 - 0.938i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.602284 + 0.862592i\)
\(L(\frac12)\) \(\approx\) \(0.602284 + 0.862592i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.49 - 1.33i)T \)
3 \( 1 - 1.73T \)
5 \( 1 \)
good7 \( 1 + 6.56T + 49T^{2} \)
11 \( 1 + 2.26iT - 121T^{2} \)
13 \( 1 - 14.8iT - 169T^{2} \)
17 \( 1 - 26.8iT - 289T^{2} \)
19 \( 1 - 10.8iT - 361T^{2} \)
23 \( 1 - 36.4T + 529T^{2} \)
29 \( 1 - 35.2T + 841T^{2} \)
31 \( 1 - 23.8iT - 961T^{2} \)
37 \( 1 - 54.7iT - 1.36e3T^{2} \)
41 \( 1 + 23.8T + 1.68e3T^{2} \)
43 \( 1 + 56.2T + 1.84e3T^{2} \)
47 \( 1 + 51.4T + 2.20e3T^{2} \)
53 \( 1 + 30.6iT - 2.80e3T^{2} \)
59 \( 1 + 6.92iT - 3.48e3T^{2} \)
61 \( 1 - 107.T + 3.72e3T^{2} \)
67 \( 1 + 111.T + 4.48e3T^{2} \)
71 \( 1 - 31.3iT - 5.04e3T^{2} \)
73 \( 1 + 110. iT - 5.32e3T^{2} \)
79 \( 1 - 59.0iT - 6.24e3T^{2} \)
83 \( 1 - 142.T + 6.88e3T^{2} \)
89 \( 1 + 7.14T + 7.92e3T^{2} \)
97 \( 1 + 126. iT - 9.40e3T^{2} \)
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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

−11.67268983115159786974873281766, −10.45131126943409323985344554473, −9.809633898999172848328489568335, −8.800824288235545704951814678487, −8.228638649404426297968419274466, −6.81868485701092896363397253457, −6.38277184785032616593342044437, −4.82974214903097136108684756868, −3.30540341196147022201835887954, −1.57214417903450694522291058764, 0.63382006096021536620316332427, 2.64141594936887003610508417274, 3.33078215386522255457621068830, 4.91728179004274843818513547755, 6.73558291501159969193281276537, 7.50648130489757938091275059769, 8.625039182356889550758575132658, 9.441535132686780957035594229284, 10.09213895900796230934505529423, 11.09796456160823529562414673793

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