Properties

Label 2-300-100.23-c1-0-27
Degree $2$
Conductor $300$
Sign $-0.401 + 0.915i$
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  + (1.04 − 0.955i)2-s + (−0.453 − 0.891i)3-s + (0.174 − 1.99i)4-s + (0.328 − 2.21i)5-s + (−1.32 − 0.495i)6-s + (3.17 + 3.17i)7-s + (−1.72 − 2.24i)8-s + (−0.587 + 0.809i)9-s + (−1.77 − 2.61i)10-s + (−1.22 − 1.68i)11-s + (−1.85 + 0.749i)12-s + (−0.339 − 2.14i)13-s + (6.34 + 0.277i)14-s + (−2.11 + 0.711i)15-s + (−3.93 − 0.694i)16-s + (−2.26 − 1.15i)17-s + ⋯
L(s)  = 1  + (0.737 − 0.675i)2-s + (−0.262 − 0.514i)3-s + (0.0871 − 0.996i)4-s + (0.146 − 0.989i)5-s + (−0.540 − 0.202i)6-s + (1.20 + 1.20i)7-s + (−0.608 − 0.793i)8-s + (−0.195 + 0.269i)9-s + (−0.560 − 0.828i)10-s + (−0.368 − 0.507i)11-s + (−0.535 + 0.216i)12-s + (−0.0942 − 0.594i)13-s + (1.69 + 0.0740i)14-s + (−0.547 + 0.183i)15-s + (−0.984 − 0.173i)16-s + (−0.548 − 0.279i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.401 + 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.401 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.401 + 0.915i$
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: \((2,\ 300,\ (\ :1/2),\ -0.401 + 0.915i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01490 - 1.55314i\)
\(L(\frac12)\) \(\approx\) \(1.01490 - 1.55314i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.04 + 0.955i)T \)
3 \( 1 + (0.453 + 0.891i)T \)
5 \( 1 + (-0.328 + 2.21i)T \)
good7 \( 1 + (-3.17 - 3.17i)T + 7iT^{2} \)
11 \( 1 + (1.22 + 1.68i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.339 + 2.14i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (2.26 + 1.15i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (-1.87 - 5.78i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.703 - 4.44i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (-1.37 - 0.445i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.68 + 0.546i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (-8.87 + 1.40i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (-5.90 - 4.28i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-4.19 + 4.19i)T - 43iT^{2} \)
47 \( 1 + (-1.82 + 0.928i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (-10.2 + 5.24i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (3.92 + 2.84i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (11.1 - 8.09i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (3.16 - 6.20i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (8.47 + 2.75i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.968 + 0.153i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (4.20 - 12.9i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (4.17 + 2.12i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (-9.07 - 12.4i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (4.76 + 9.35i)T + (-57.0 + 78.4i)T^{2} \)
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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

−11.76195151212150846090260603644, −10.90164086182657284379203776363, −9.635836847424535828839322816846, −8.611187006372270403997276028990, −7.74116050190525270627076409937, −5.74539658181843898939797232058, −5.58396116278930935660697408779, −4.39259067902588226139312415267, −2.56112350915124015019799914893, −1.33371980568565018263108966349, 2.59967627653891464628719793952, 4.22907234819706105578800759493, 4.70466642342085347978875890787, 6.16741325783901443179101763676, 7.14020463625802037621352880675, 7.78585625452852022931833657013, 9.146641544562088391722528618069, 10.53394457517135791777037439968, 11.06605006444756878786713352470, 11.89770117830197007496923725336

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