Properties

Label 2-300-100.67-c1-0-14
Degree $2$
Conductor $300$
Sign $0.661 - 0.749i$
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  + (0.300 + 1.38i)2-s + (−0.987 − 0.156i)3-s + (−1.81 + 0.830i)4-s + (2.22 − 0.222i)5-s + (−0.0804 − 1.41i)6-s + (3.70 − 3.70i)7-s + (−1.69 − 2.26i)8-s + (0.951 + 0.309i)9-s + (0.976 + 3.00i)10-s + (−0.352 + 0.114i)11-s + (1.92 − 0.535i)12-s + (2.36 + 4.64i)13-s + (6.22 + 4.00i)14-s + (−2.23 − 0.128i)15-s + (2.62 − 3.02i)16-s + (−0.155 − 0.980i)17-s + ⋯
L(s)  = 1  + (0.212 + 0.977i)2-s + (−0.570 − 0.0903i)3-s + (−0.909 + 0.415i)4-s + (0.995 − 0.0995i)5-s + (−0.0328 − 0.576i)6-s + (1.39 − 1.39i)7-s + (−0.598 − 0.800i)8-s + (0.317 + 0.103i)9-s + (0.308 + 0.951i)10-s + (−0.106 + 0.0344i)11-s + (0.556 − 0.154i)12-s + (0.655 + 1.28i)13-s + (1.66 + 1.07i)14-s + (−0.576 − 0.0330i)15-s + (0.655 − 0.755i)16-s + (−0.0376 − 0.237i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.661 - 0.749i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.661 - 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31929 + 0.595055i\)
\(L(\frac12)\) \(\approx\) \(1.31929 + 0.595055i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.300 - 1.38i)T \)
3 \( 1 + (0.987 + 0.156i)T \)
5 \( 1 + (-2.22 + 0.222i)T \)
good7 \( 1 + (-3.70 + 3.70i)T - 7iT^{2} \)
11 \( 1 + (0.352 - 0.114i)T + (8.89 - 6.46i)T^{2} \)
13 \( 1 + (-2.36 - 4.64i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (0.155 + 0.980i)T + (-16.1 + 5.25i)T^{2} \)
19 \( 1 + (2.30 + 1.67i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (3.21 - 6.30i)T + (-13.5 - 18.6i)T^{2} \)
29 \( 1 + (-0.959 - 1.32i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (-3.91 + 5.39i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (3.78 - 1.92i)T + (21.7 - 29.9i)T^{2} \)
41 \( 1 + (-0.959 + 2.95i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (-0.302 - 0.302i)T + 43iT^{2} \)
47 \( 1 + (0.817 - 5.16i)T + (-44.6 - 14.5i)T^{2} \)
53 \( 1 + (0.685 - 4.32i)T + (-50.4 - 16.3i)T^{2} \)
59 \( 1 + (1.32 - 4.07i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.66 + 8.20i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (4.13 - 0.654i)T + (63.7 - 20.7i)T^{2} \)
71 \( 1 + (3.73 + 5.14i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (12.0 + 6.12i)T + (42.9 + 59.0i)T^{2} \)
79 \( 1 + (7.17 - 5.21i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-1.07 - 6.78i)T + (-78.9 + 25.6i)T^{2} \)
89 \( 1 + (12.1 - 3.93i)T + (72.0 - 52.3i)T^{2} \)
97 \( 1 + (-10.9 - 1.72i)T + (92.2 + 29.9i)T^{2} \)
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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

−11.82718892740137724471797163101, −10.94080584523376976126834797266, −9.948873542153235600124872903635, −8.896491894790311105349553885737, −7.77741788919048438122771278433, −6.92121000235424483682572616991, −5.99042082784170046888031739769, −4.86071977269459146685385571083, −4.13529757965155936681691318662, −1.46578295708219410109913006231, 1.59883520215873560931594326768, 2.74026994121004545261467682972, 4.60621419741414961580153417864, 5.51774854737775525190407774537, 6.08950224652104785119559694807, 8.299754505393630057218267942874, 8.775637227924613254931583671889, 10.20109197821596109177531290646, 10.63469598069082461303509235535, 11.63149458211146907777728929536

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