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Properties

Label 2-300-60.23-c2-0-34
Degree $2$
Conductor $300$
Sign $0.997 + 0.0643i$
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  + (0.141 + 1.99i)2-s + (−2.06 + 2.17i)3-s + (−3.95 + 0.565i)4-s + (−4.63 − 3.81i)6-s + (−5.18 − 5.18i)7-s + (−1.68 − 7.81i)8-s + (−0.459 − 8.98i)9-s + 7.14·11-s + (6.95 − 9.78i)12-s + (7.93 + 7.93i)13-s + (9.61 − 11.0i)14-s + (15.3 − 4.47i)16-s + (−16.5 − 16.5i)17-s + (17.8 − 2.19i)18-s + 12.1·19-s + ⋯
L(s)  = 1  + (0.0708 + 0.997i)2-s + (−0.688 + 0.724i)3-s + (−0.989 + 0.141i)4-s + (−0.771 − 0.635i)6-s + (−0.741 − 0.741i)7-s + (−0.211 − 0.977i)8-s + (−0.0510 − 0.998i)9-s + 0.649·11-s + (0.579 − 0.815i)12-s + (0.610 + 0.610i)13-s + (0.686 − 0.791i)14-s + (0.960 − 0.279i)16-s + (−0.975 − 0.975i)17-s + (0.992 − 0.121i)18-s + 0.639·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0643i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0643i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.997 + 0.0643i$
Analytic conductor: \(8.17440\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1),\ 0.997 + 0.0643i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.730078 - 0.0235245i\)
\(L(\frac12)\) \(\approx\) \(0.730078 - 0.0235245i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.141 - 1.99i)T \)
3 \( 1 + (2.06 - 2.17i)T \)
5 \( 1 \)
good7 \( 1 + (5.18 + 5.18i)T + 49iT^{2} \)
11 \( 1 - 7.14T + 121T^{2} \)
13 \( 1 + (-7.93 - 7.93i)T + 169iT^{2} \)
17 \( 1 + (16.5 + 16.5i)T + 289iT^{2} \)
19 \( 1 - 12.1T + 361T^{2} \)
23 \( 1 + (11.0 + 11.0i)T + 529iT^{2} \)
29 \( 1 + 26.1T + 841T^{2} \)
31 \( 1 - 8.74iT - 961T^{2} \)
37 \( 1 + (-26.7 + 26.7i)T - 1.36e3iT^{2} \)
41 \( 1 + 35.4iT - 1.68e3T^{2} \)
43 \( 1 + (-24.6 + 24.6i)T - 1.84e3iT^{2} \)
47 \( 1 + (-58.6 + 58.6i)T - 2.20e3iT^{2} \)
53 \( 1 + (20.4 - 20.4i)T - 2.80e3iT^{2} \)
59 \( 1 + 59.4iT - 3.48e3T^{2} \)
61 \( 1 - 7.42T + 3.72e3T^{2} \)
67 \( 1 + (35.8 + 35.8i)T + 4.48e3iT^{2} \)
71 \( 1 + 46.2T + 5.04e3T^{2} \)
73 \( 1 + (10.6 + 10.6i)T + 5.32e3iT^{2} \)
79 \( 1 - 68.3T + 6.24e3T^{2} \)
83 \( 1 + (76.6 + 76.6i)T + 6.88e3iT^{2} \)
89 \( 1 + 41.0T + 7.92e3T^{2} \)
97 \( 1 + (81.7 - 81.7i)T - 9.40e3iT^{2} \)
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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

−11.46417200039525725120773506801, −10.42802617373787087598754426727, −9.420225807707449609162232495200, −8.920447831265456909378673836439, −7.25203535655385500281851300621, −6.59580889767008732709117114036, −5.63931395710184837340645672253, −4.37948064631665579530843705714, −3.66397659214282593117176567092, −0.42445668269915607082038985152, 1.33524395479763455735448302404, 2.74931241335577647906030948281, 4.17192922737388194703337218585, 5.67000787957184845737431861850, 6.26387014953959984449083815577, 7.81455401148328034075630934292, 8.873732991289131002789300333331, 9.792020332074023802540352285251, 10.91292843718090862426582989696, 11.54006642356234177675136647731

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