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Properties

Label 2-300-300.119-c1-0-39
Degree $2$
Conductor $300$
Sign $0.445 + 0.895i$
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  + (1.29 − 0.576i)2-s + (−1.32 − 1.11i)3-s + (1.33 − 1.48i)4-s + (1.53 + 1.62i)5-s + (−2.35 − 0.667i)6-s + 2.22·7-s + (0.866 − 2.69i)8-s + (0.534 + 2.95i)9-s + (2.91 + 1.21i)10-s + (−0.989 − 3.04i)11-s + (−3.42 + 0.496i)12-s + (2.10 + 0.683i)13-s + (2.86 − 1.27i)14-s + (−0.231 − 3.86i)15-s + (−0.433 − 3.97i)16-s + (−6.14 − 4.46i)17-s + ⋯
L(s)  = 1  + (0.913 − 0.407i)2-s + (−0.767 − 0.641i)3-s + (0.667 − 0.744i)4-s + (0.685 + 0.727i)5-s + (−0.962 − 0.272i)6-s + 0.839·7-s + (0.306 − 0.951i)8-s + (0.178 + 0.984i)9-s + (0.922 + 0.385i)10-s + (−0.298 − 0.917i)11-s + (−0.989 + 0.143i)12-s + (0.583 + 0.189i)13-s + (0.766 − 0.342i)14-s + (−0.0597 − 0.998i)15-s + (−0.108 − 0.994i)16-s + (−1.49 − 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.445 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.445 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.445 + 0.895i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.445 + 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70369 - 1.05564i\)
\(L(\frac12)\) \(\approx\) \(1.70369 - 1.05564i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.29 + 0.576i)T \)
3 \( 1 + (1.32 + 1.11i)T \)
5 \( 1 + (-1.53 - 1.62i)T \)
good7 \( 1 - 2.22T + 7T^{2} \)
11 \( 1 + (0.989 + 3.04i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-2.10 - 0.683i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (6.14 + 4.46i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.90 - 2.62i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (0.789 - 0.256i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (-5.45 - 7.51i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (2.79 - 3.84i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (-6.83 - 2.21i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (6.82 + 2.21i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 - 2.43T + 43T^{2} \)
47 \( 1 + (-2.58 - 3.55i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-3.58 + 2.60i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.60 - 11.0i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (0.275 + 0.847i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (8.14 + 5.91i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (10.5 - 7.65i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-2.71 + 0.880i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-0.510 - 0.702i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (3.39 - 4.67i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + (-2.71 + 0.882i)T + (72.0 - 52.3i)T^{2} \)
97 \( 1 + (4.87 + 6.71i)T + (-29.9 + 92.2i)T^{2} \)
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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

−11.39625018714156556481499149274, −11.02321944409070140881816306802, −10.29421050858045073148249859379, −8.707250868415183757774165443739, −7.24123824384752273931425478349, −6.43044504118654491299682116971, −5.60115523890240969507348598724, −4.60429487862405269761428535530, −2.84038658956138915620655800100, −1.57727153890590367110863923775, 2.08506275507086845495993462122, 4.28097348859695308513944664738, 4.68118708049088673918780120831, 5.82549924978770863068798175250, 6.55812478791115542278530647914, 8.048566343689583716329978944755, 9.003338595005399371660347919176, 10.29807637444665264238721721689, 11.11702692665289476056751552413, 11.96932973518607888390948845566

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