↑

Properties

Label 2-300-100.23-c1-0-16
Degree $2$
Conductor $300$
Sign $0.161 - 0.986i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.16 + 0.808i)2-s + (0.453 + 0.891i)3-s + (0.692 + 1.87i)4-s + (2.17 + 0.536i)5-s + (−0.193 + 1.40i)6-s + (−0.403 − 0.403i)7-s + (−0.712 + 2.73i)8-s + (−0.587 + 0.809i)9-s + (2.08 + 2.37i)10-s + (−2.68 − 3.69i)11-s + (−1.35 + 1.46i)12-s + (−0.669 − 4.22i)13-s + (−0.141 − 0.793i)14-s + (0.507 + 2.17i)15-s + (−3.03 + 2.59i)16-s + (1.13 + 0.576i)17-s + ⋯
L(s)  = 1  + (0.820 + 0.571i)2-s + (0.262 + 0.514i)3-s + (0.346 + 0.938i)4-s + (0.970 + 0.240i)5-s + (−0.0790 + 0.571i)6-s + (−0.152 − 0.152i)7-s + (−0.252 + 0.967i)8-s + (−0.195 + 0.269i)9-s + (0.659 + 0.751i)10-s + (−0.809 − 1.11i)11-s + (−0.391 + 0.424i)12-s + (−0.185 − 1.17i)13-s + (−0.0379 − 0.212i)14-s + (0.130 + 0.562i)15-s + (−0.759 + 0.649i)16-s + (0.274 + 0.139i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.161 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.161 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.161 - 0.986i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.161 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76447 + 1.49853i\)
\(L(\frac12)\) \(\approx\) \(1.76447 + 1.49853i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.16 - 0.808i)T \)
3 \( 1 + (-0.453 - 0.891i)T \)
5 \( 1 + (-2.17 - 0.536i)T \)
good7 \( 1 + (0.403 + 0.403i)T + 7iT^{2} \)
11 \( 1 + (2.68 + 3.69i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.669 + 4.22i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-1.13 - 0.576i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (0.214 + 0.658i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.103 - 0.654i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (-5.48 - 1.78i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (7.87 - 2.55i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.551 - 0.0872i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (3.58 + 2.60i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (2.72 - 2.72i)T - 43iT^{2} \)
47 \( 1 + (-9.06 + 4.61i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (-7.66 + 3.90i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (-5.72 - 4.16i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (10.7 - 7.80i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-4.20 + 8.25i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (1.71 + 0.555i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (6.95 + 1.10i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (2.36 - 7.28i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-7.97 - 4.06i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (-3.05 - 4.19i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (-5.74 - 11.2i)T + (-57.0 + 78.4i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.19410448136280687725880115676, −10.80399578654637081037433665674, −10.32826056955358010461360489269, −8.941860155202024446494416361731, −8.090777186633181239855840024302, −6.92713241303641259748126929101, −5.67248843644805926992477968447, −5.22935787696226234912196719240, −3.54193688327424938136693853206, −2.65628961611517739906023832665, 1.75041746311678302973501454177, 2.64910379279981846464383621415, 4.36649173340539109873469286134, 5.41377713962837336130741611453, 6.44903393072748775514049687169, 7.38822267939279005093162429973, 9.008269605425148945487625033643, 9.773373244778301865895609705312, 10.59906099018533285021716661305, 11.88002267862150235385389777506

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