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Properties

Label 2-300-25.9-c1-0-1
Degree $2$
Conductor $300$
Sign $0.920 - 0.390i$
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  + (−0.951 + 0.309i)3-s + (0.913 + 2.04i)5-s − 4.62i·7-s + (0.809 − 0.587i)9-s + (4.00 + 2.90i)11-s + (2.21 + 3.04i)13-s + (−1.49 − 1.65i)15-s + (2.55 + 0.831i)17-s + (−1.81 + 5.58i)19-s + (1.42 + 4.40i)21-s + (3.92 − 5.40i)23-s + (−3.33 + 3.72i)25-s + (−0.587 + 0.809i)27-s + (−0.370 − 1.14i)29-s + (1.02 − 3.14i)31-s + ⋯
L(s)  = 1  + (−0.549 + 0.178i)3-s + (0.408 + 0.912i)5-s − 1.74i·7-s + (0.269 − 0.195i)9-s + (1.20 + 0.877i)11-s + (0.613 + 0.844i)13-s + (−0.387 − 0.428i)15-s + (0.620 + 0.201i)17-s + (−0.416 + 1.28i)19-s + (0.311 + 0.960i)21-s + (0.818 − 1.12i)23-s + (−0.666 + 0.745i)25-s + (−0.113 + 0.155i)27-s + (−0.0688 − 0.212i)29-s + (0.183 − 0.564i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.390i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 - 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.920 - 0.390i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.920 - 0.390i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23136 + 0.250647i\)
\(L(\frac12)\) \(\approx\) \(1.23136 + 0.250647i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.951 - 0.309i)T \)
5 \( 1 + (-0.913 - 2.04i)T \)
good7 \( 1 + 4.62iT - 7T^{2} \)
11 \( 1 + (-4.00 - 2.90i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-2.21 - 3.04i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.55 - 0.831i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.81 - 5.58i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-3.92 + 5.40i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.370 + 1.14i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.02 + 3.14i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.10 - 1.51i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-2.45 + 1.78i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 10.6iT - 43T^{2} \)
47 \( 1 + (-0.246 + 0.0801i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (9.31 - 3.02i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (7.78 - 5.65i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (5.07 + 3.68i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (2.43 + 0.791i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (-2.68 - 8.25i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-2.86 + 3.94i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.85 + 11.8i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (8.45 + 2.74i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (-11.7 - 8.56i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (3.79 - 1.23i)T + (78.4 - 57.0i)T^{2} \)
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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

−11.63214659913176169937768909763, −10.66973355510358697546751083420, −10.23232787102634668816450913217, −9.265988060312182795788589842041, −7.63915373217284125737325122452, −6.78539953150197892534115242972, −6.18554157248306668362186703528, −4.39628149755638001367974066981, −3.70523753546853871226956318048, −1.51678797778902104482634868763, 1.29158744293026687825423424852, 3.06612245332385136939377401396, 4.90046454411962630918999726625, 5.72266333874601415028523703809, 6.38963537228029993259853950960, 8.087730240106636338133383506863, 8.996233447004163164927449463371, 9.454310286393963960767649322257, 11.08554737753911990583710938949, 11.68637024089753144226408049730

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