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Properties

Label 2-300-100.27-c1-0-13
Degree $2$
Conductor $300$
Sign $0.999 - 0.0183i$
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  + (1.33 − 0.472i)2-s + (−0.891 + 0.453i)3-s + (1.55 − 1.25i)4-s + (1.01 + 1.99i)5-s + (−0.973 + 1.02i)6-s + (0.189 − 0.189i)7-s + (1.47 − 2.41i)8-s + (0.587 − 0.809i)9-s + (2.29 + 2.17i)10-s + (2.68 + 3.69i)11-s + (−0.812 + 1.82i)12-s + (−1.66 + 0.263i)13-s + (0.163 − 0.342i)14-s + (−1.80 − 1.31i)15-s + (0.828 − 3.91i)16-s + (2.52 − 4.95i)17-s + ⋯
L(s)  = 1  + (0.942 − 0.333i)2-s + (−0.514 + 0.262i)3-s + (0.776 − 0.629i)4-s + (0.452 + 0.891i)5-s + (−0.397 + 0.418i)6-s + (0.0717 − 0.0717i)7-s + (0.522 − 0.852i)8-s + (0.195 − 0.269i)9-s + (0.724 + 0.689i)10-s + (0.808 + 1.11i)11-s + (−0.234 + 0.527i)12-s + (−0.462 + 0.0731i)13-s + (0.0436 − 0.0915i)14-s + (−0.466 − 0.340i)15-s + (0.207 − 0.978i)16-s + (0.611 − 1.20i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0183i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0183i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.999 - 0.0183i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.999 - 0.0183i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.09375 + 0.0192113i\)
\(L(\frac12)\) \(\approx\) \(2.09375 + 0.0192113i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.33 + 0.472i)T \)
3 \( 1 + (0.891 - 0.453i)T \)
5 \( 1 + (-1.01 - 1.99i)T \)
good7 \( 1 + (-0.189 + 0.189i)T - 7iT^{2} \)
11 \( 1 + (-2.68 - 3.69i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (1.66 - 0.263i)T + (12.3 - 4.01i)T^{2} \)
17 \( 1 + (-2.52 + 4.95i)T + (-9.99 - 13.7i)T^{2} \)
19 \( 1 + (0.186 + 0.574i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (3.67 + 0.581i)T + (21.8 + 7.10i)T^{2} \)
29 \( 1 + (-3.14 - 1.02i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (6.47 - 2.10i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.372 + 2.35i)T + (-35.1 + 11.4i)T^{2} \)
41 \( 1 + (9.30 + 6.76i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-0.00980 - 0.00980i)T + 43iT^{2} \)
47 \( 1 + (1.17 + 2.31i)T + (-27.6 + 38.0i)T^{2} \)
53 \( 1 + (4.53 + 8.89i)T + (-31.1 + 42.8i)T^{2} \)
59 \( 1 + (-3.15 - 2.29i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (9.40 - 6.83i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (6.49 + 3.30i)T + (39.3 + 54.2i)T^{2} \)
71 \( 1 + (-12.6 - 4.09i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.770 - 4.86i)T + (-69.4 - 22.5i)T^{2} \)
79 \( 1 + (-4.17 + 12.8i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (3.42 - 6.72i)T + (-48.7 - 67.1i)T^{2} \)
89 \( 1 + (-4.20 - 5.78i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (0.917 - 0.467i)T + (57.0 - 78.4i)T^{2} \)
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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

−11.89226712250151306276149785889, −10.92705710494336995064850979108, −10.07126602143981105767447275208, −9.456753879696228798638628567648, −7.29319501371539943111898220055, −6.76820706159290407196838688545, −5.60926746588969588490660937658, −4.63284482831721960809009448779, −3.39264288156590478348342924653, −1.96554194397238350990804277236, 1.67394581514406536289974816778, 3.58032899241007717443825845386, 4.79808628670086740297392275250, 5.83196721817750222205306545404, 6.36882560552128221208520100877, 7.83630286714408935051368940875, 8.624396951468068564541425058943, 9.995475655358149810052775238567, 11.15388274078272252978656911604, 12.03483285136731522522639842469

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