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Properties

Label 2-300-60.59-c1-0-24
Degree $2$
Conductor $300$
Sign $0.875 - 0.483i$
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  + (1.28 + 0.599i)2-s + (1.66 − 0.468i)3-s + (1.28 + 1.53i)4-s + (2.41 + 0.400i)6-s − 0.936·7-s + (0.719 + 2.73i)8-s + (2.56 − 1.56i)9-s − 4.27·11-s + (2.85 + 1.96i)12-s − 3.12i·13-s + (−1.19 − 0.561i)14-s + (−0.719 + 3.93i)16-s − 2·17-s + (4.21 − 0.463i)18-s + 4.27i·19-s + ⋯
L(s)  = 1  + (0.905 + 0.424i)2-s + (0.962 − 0.270i)3-s + (0.640 + 0.768i)4-s + (0.986 + 0.163i)6-s − 0.353·7-s + (0.254 + 0.967i)8-s + (0.853 − 0.520i)9-s − 1.28·11-s + (0.824 + 0.566i)12-s − 0.866i·13-s + (−0.320 − 0.150i)14-s + (−0.179 + 0.983i)16-s − 0.485·17-s + (0.994 − 0.109i)18-s + 0.979i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 - 0.483i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.875 - 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.875 - 0.483i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.875 - 0.483i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.56232 + 0.661211i\)
\(L(\frac12)\) \(\approx\) \(2.56232 + 0.661211i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.28 - 0.599i)T \)
3 \( 1 + (-1.66 + 0.468i)T \)
5 \( 1 \)
good7 \( 1 + 0.936T + 7T^{2} \)
11 \( 1 + 4.27T + 11T^{2} \)
13 \( 1 + 3.12iT - 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 4.27iT - 19T^{2} \)
23 \( 1 + 7.60iT - 23T^{2} \)
29 \( 1 - 5.12iT - 29T^{2} \)
31 \( 1 + 2.39iT - 31T^{2} \)
37 \( 1 + 3.12iT - 37T^{2} \)
41 \( 1 - 7.12iT - 41T^{2} \)
43 \( 1 - 1.46T + 43T^{2} \)
47 \( 1 + 0.936iT - 47T^{2} \)
53 \( 1 - 4.24T + 53T^{2} \)
59 \( 1 - 7.19T + 59T^{2} \)
61 \( 1 + 5.12T + 61T^{2} \)
67 \( 1 + 5.20T + 67T^{2} \)
71 \( 1 - 6.67T + 71T^{2} \)
73 \( 1 - 8.24iT - 73T^{2} \)
79 \( 1 + 9.06iT - 79T^{2} \)
83 \( 1 - 4.68iT - 83T^{2} \)
89 \( 1 - 6.24iT - 89T^{2} \)
97 \( 1 + 6iT - 97T^{2} \)
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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

−12.40899137443446028492979818768, −10.88496300443369316536086823613, −10.01847134508176820449332796813, −8.543309544780603672278905884271, −7.941649147637576721726949880687, −6.97787043199817082966856775787, −5.87534350301891697550193613195, −4.62041650311635729780935191155, −3.33195707669791873931256695076, −2.39849004701654073599190469424, 2.10971197066226745619528438226, 3.15858736582851018107682320876, 4.32097337955188654482571814383, 5.33861784205783069771007118629, 6.77543910182578644308898838301, 7.70235677392549845145314095871, 9.065087473140376063948904252416, 9.867897376675035580056677581525, 10.77426482509852337946012285360, 11.70635896202546437086121982292

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