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Properties

Label 2-300-25.9-c1-0-0
Degree $2$
Conductor $300$
Sign $0.496 - 0.868i$
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  + (0.951 − 0.309i)3-s + (−1.64 + 1.51i)5-s + 3.78i·7-s + (0.809 − 0.587i)9-s + (0.653 + 0.474i)11-s + (2.79 + 3.84i)13-s + (−1.09 + 1.95i)15-s + (−1.09 − 0.355i)17-s + (−0.00463 + 0.0142i)19-s + (1.17 + 3.60i)21-s + (3.68 − 5.07i)23-s + (0.395 − 4.98i)25-s + (0.587 − 0.809i)27-s + (−1.14 − 3.51i)29-s + (−0.488 + 1.50i)31-s + ⋯
L(s)  = 1  + (0.549 − 0.178i)3-s + (−0.734 + 0.678i)5-s + 1.43i·7-s + (0.269 − 0.195i)9-s + (0.197 + 0.143i)11-s + (0.774 + 1.06i)13-s + (−0.282 + 0.503i)15-s + (−0.264 − 0.0861i)17-s + (−0.00106 + 0.00327i)19-s + (0.255 + 0.786i)21-s + (0.768 − 1.05i)23-s + (0.0790 − 0.996i)25-s + (0.113 − 0.155i)27-s + (−0.212 − 0.653i)29-s + (−0.0878 + 0.270i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.496 - 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.496 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.496 - 0.868i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.496 - 0.868i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18074 + 0.685335i\)
\(L(\frac12)\) \(\approx\) \(1.18074 + 0.685335i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.951 + 0.309i)T \)
5 \( 1 + (1.64 - 1.51i)T \)
good7 \( 1 - 3.78iT - 7T^{2} \)
11 \( 1 + (-0.653 - 0.474i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-2.79 - 3.84i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.09 + 0.355i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.00463 - 0.0142i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-3.68 + 5.07i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (1.14 + 3.51i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.488 - 1.50i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-5.02 - 6.91i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (9.30 - 6.75i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 10.2iT - 43T^{2} \)
47 \( 1 + (-0.500 + 0.162i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (2.80 - 0.911i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-9.25 + 6.72i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (2.54 + 1.84i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-12.6 - 4.10i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (1.51 + 4.67i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-2.75 + 3.78i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.86 + 8.81i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (1.35 + 0.439i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (13.0 + 9.46i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (7.66 - 2.49i)T + (78.4 - 57.0i)T^{2} \)
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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

−11.78441950507315464546922438544, −11.24803074334097971853391350671, −9.916490971260867746718392440055, −8.806156029299838631662656298877, −8.332626704258982482632946099101, −6.96964064014825991802955267986, −6.23310047461744925884079387481, −4.63727425366019339264523641583, −3.33810073764607272107565668718, −2.18067387694999681322047917812, 1.05759334084503902380715569559, 3.40596476388928154901559020967, 4.11645782578869651777328255987, 5.37420454977508109414860600433, 7.01682807317215518278052331771, 7.81122193984947823031898854639, 8.623886910564537810170765832651, 9.662494991010611998914483710210, 10.74677004766860986591196144363, 11.36923830959154735620382933018

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