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Properties

Label 2-300-60.23-c2-0-59
Degree $2$
Conductor $300$
Sign $-0.996 + 0.0792i$
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  + (1.68 − 1.07i)2-s + (−2.99 − 0.130i)3-s + (1.67 − 3.63i)4-s + (−5.18 + 3.01i)6-s + (−1.91 − 1.91i)7-s + (−1.11 − 7.92i)8-s + (8.96 + 0.782i)9-s − 6.87·11-s + (−5.47 + 10.6i)12-s + (−12.2 − 12.2i)13-s + (−5.29 − 1.15i)14-s + (−10.4 − 12.1i)16-s + (9.47 + 9.47i)17-s + (15.9 − 8.36i)18-s − 33.2·19-s + ⋯
L(s)  = 1  + (0.841 − 0.539i)2-s + (−0.999 − 0.0434i)3-s + (0.417 − 0.908i)4-s + (−0.864 + 0.502i)6-s + (−0.273 − 0.273i)7-s + (−0.138 − 0.990i)8-s + (0.996 + 0.0869i)9-s − 0.624·11-s + (−0.456 + 0.889i)12-s + (−0.944 − 0.944i)13-s + (−0.378 − 0.0826i)14-s + (−0.651 − 0.758i)16-s + (0.557 + 0.557i)17-s + (0.885 − 0.464i)18-s − 1.75·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0792i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.996 + 0.0792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.996 + 0.0792i$
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: \((2,\ 300,\ (\ :1),\ -0.996 + 0.0792i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0405303 - 1.02080i\)
\(L(\frac12)\) \(\approx\) \(0.0405303 - 1.02080i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.68 + 1.07i)T \)
3 \( 1 + (2.99 + 0.130i)T \)
5 \( 1 \)
good7 \( 1 + (1.91 + 1.91i)T + 49iT^{2} \)
11 \( 1 + 6.87T + 121T^{2} \)
13 \( 1 + (12.2 + 12.2i)T + 169iT^{2} \)
17 \( 1 + (-9.47 - 9.47i)T + 289iT^{2} \)
19 \( 1 + 33.2T + 361T^{2} \)
23 \( 1 + (7.20 + 7.20i)T + 529iT^{2} \)
29 \( 1 + 2.29T + 841T^{2} \)
31 \( 1 + 12.1iT - 961T^{2} \)
37 \( 1 + (-20.7 + 20.7i)T - 1.36e3iT^{2} \)
41 \( 1 - 50.9iT - 1.68e3T^{2} \)
43 \( 1 + (-15.1 + 15.1i)T - 1.84e3iT^{2} \)
47 \( 1 + (-26.7 + 26.7i)T - 2.20e3iT^{2} \)
53 \( 1 + (-15.5 + 15.5i)T - 2.80e3iT^{2} \)
59 \( 1 + 63.0iT - 3.48e3T^{2} \)
61 \( 1 - 28.4T + 3.72e3T^{2} \)
67 \( 1 + (-32.4 - 32.4i)T + 4.48e3iT^{2} \)
71 \( 1 - 88.8T + 5.04e3T^{2} \)
73 \( 1 + (71.1 + 71.1i)T + 5.32e3iT^{2} \)
79 \( 1 - 75.1T + 6.24e3T^{2} \)
83 \( 1 + (-58.6 - 58.6i)T + 6.88e3iT^{2} \)
89 \( 1 - 41.1T + 7.92e3T^{2} \)
97 \( 1 + (30.3 - 30.3i)T - 9.40e3iT^{2} \)
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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

−11.03829181291669914406069205835, −10.41315394958453445176321871730, −9.784653086311490835801370510791, −7.953116496298268784738142129909, −6.77186891631554524079719356519, −5.85785072537450901336755299508, −4.96288276769726272442940258818, −3.88755815377869850734283041722, −2.27645862410955702765793721535, −0.39819374128145893618338505154, 2.38091472074427053478046769695, 4.11588773880459085185855279531, 5.01143942687457429716770910279, 5.98376390916170962680773486348, 6.85790169302338433044547511229, 7.74705851662276622326727440573, 9.136823469338724181388897635098, 10.33798668761881524655730493467, 11.29617897186500623028500696212, 12.21769485271130867280932703697

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