Properties

Label 2-300-12.11-c1-0-21
Degree $2$
Conductor $300$
Sign $0.997 - 0.0642i$
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  + (−0.273 + 1.38i)2-s + (1.55 − 0.758i)3-s + (−1.85 − 0.758i)4-s + (0.626 + 2.36i)6-s − 3.56i·7-s + (1.55 − 2.36i)8-s + (1.85 − 2.36i)9-s + 4.20·11-s + (−3.45 + 0.222i)12-s − 2.70·13-s + (4.94 + 0.973i)14-s + (2.85 + 2.80i)16-s − 0.828i·17-s + (2.77 + 3.21i)18-s + 5.07i·19-s + ⋯
L(s)  = 1  + (−0.193 + 0.981i)2-s + (0.899 − 0.437i)3-s + (−0.925 − 0.379i)4-s + (0.255 + 0.966i)6-s − 1.34i·7-s + (0.550 − 0.834i)8-s + (0.616 − 0.787i)9-s + 1.26·11-s + (−0.997 + 0.0642i)12-s − 0.749·13-s + (1.32 + 0.260i)14-s + (0.712 + 0.701i)16-s − 0.200i·17-s + (0.653 + 0.757i)18-s + 1.16i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.997 - 0.0642i$
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: \((2,\ 300,\ (\ :1/2),\ 0.997 - 0.0642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47214 + 0.0473093i\)
\(L(\frac12)\) \(\approx\) \(1.47214 + 0.0473093i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.273 - 1.38i)T \)
3 \( 1 + (-1.55 + 0.758i)T \)
5 \( 1 \)
good7 \( 1 + 3.56iT - 7T^{2} \)
11 \( 1 - 4.20T + 11T^{2} \)
13 \( 1 + 2.70T + 13T^{2} \)
17 \( 1 + 0.828iT - 17T^{2} \)
19 \( 1 - 5.07iT - 19T^{2} \)
23 \( 1 - 1.09T + 23T^{2} \)
29 \( 1 - 5.55iT - 29T^{2} \)
31 \( 1 + 6.59iT - 31T^{2} \)
37 \( 1 + 5.40T + 37T^{2} \)
41 \( 1 - 10.2iT - 41T^{2} \)
43 \( 1 + 0.531iT - 43T^{2} \)
47 \( 1 - 6.22T + 47T^{2} \)
53 \( 1 - 5.55iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 0.701T + 61T^{2} \)
67 \( 1 - 2.04iT - 67T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
73 \( 1 + 7.70T + 73T^{2} \)
79 \( 1 + 7.12iT - 79T^{2} \)
83 \( 1 - 3.11T + 83T^{2} \)
89 \( 1 + 4.72iT - 89T^{2} \)
97 \( 1 - 8.10T + 97T^{2} \)
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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

−11.97879532836003253206343100916, −10.40392638506841382307264572865, −9.627513437910253680292356035274, −8.779369670688478844692031082306, −7.66081807381551672209459476264, −7.15388393734582442485739710632, −6.20143834921349963789234930570, −4.48230417326699697896028262777, −3.60875232184222763226009836933, −1.27566324601648472071922053583, 2.00401386698956217742619062919, 3.01421242577524037355828365221, 4.25208470034793145585596531315, 5.34629657538372706406771740101, 7.10910356955597002090270779288, 8.516571127836937985270028297812, 9.005913620723219427754164032201, 9.668987800330738455242694856007, 10.73907078713133835753385634698, 11.84138204935040754825141234835

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