Properties

Label 2-300-25.11-c1-0-1
Degree $2$
Conductor $300$
Sign $0.971 + 0.238i$
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  + (−0.309 + 0.951i)3-s + (−0.962 − 2.01i)5-s + 1.50·7-s + (−0.809 − 0.587i)9-s + (4.99 − 3.62i)11-s + (2.87 + 2.09i)13-s + (2.21 − 0.291i)15-s + (0.153 + 0.471i)17-s + (0.0963 + 0.296i)19-s + (−0.464 + 1.43i)21-s + (2.47 − 1.79i)23-s + (−3.14 + 3.88i)25-s + (0.809 − 0.587i)27-s + (−0.0378 + 0.116i)29-s + (−0.909 − 2.79i)31-s + ⋯
L(s)  = 1  + (−0.178 + 0.549i)3-s + (−0.430 − 0.902i)5-s + 0.568·7-s + (−0.269 − 0.195i)9-s + (1.50 − 1.09i)11-s + (0.797 + 0.579i)13-s + (0.572 − 0.0752i)15-s + (0.0371 + 0.114i)17-s + (0.0220 + 0.0680i)19-s + (−0.101 + 0.312i)21-s + (0.516 − 0.375i)23-s + (−0.629 + 0.776i)25-s + (0.155 − 0.113i)27-s + (−0.00701 + 0.0216i)29-s + (−0.163 − 0.502i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.238i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 + 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.971 + 0.238i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.971 + 0.238i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28147 - 0.155369i\)
\(L(\frac12)\) \(\approx\) \(1.28147 - 0.155369i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (0.962 + 2.01i)T \)
good7 \( 1 - 1.50T + 7T^{2} \)
11 \( 1 + (-4.99 + 3.62i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-2.87 - 2.09i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.153 - 0.471i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-0.0963 - 0.296i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-2.47 + 1.79i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.0378 - 0.116i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.909 + 2.79i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-3.53 - 2.56i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (3.44 + 2.50i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 3.62T + 43T^{2} \)
47 \( 1 + (-1.63 + 5.02i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (2.65 - 8.17i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (10.4 + 7.57i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-9.15 + 6.65i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-4.09 - 12.5i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-1.00 + 3.10i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (12.9 - 9.38i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.63 - 8.11i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-3.50 - 10.7i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (11.4 - 8.30i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (3.54 - 10.9i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.41309233192156619710112938142, −11.19720031395466236972789333755, −9.652105669606410374060282354924, −8.786563332554231157861622576675, −8.263916904581235513758671327254, −6.68043767129828985508832573031, −5.61416475042511061720528974974, −4.42214207991710728987637383563, −3.62112097583148634819482119417, −1.23843541769502200255114355943, 1.60832468618487252793367087971, 3.30127038855072042878500478438, 4.56682582474784768637955087125, 6.08876086115061994926507196525, 6.95648504927111163067587597195, 7.72928546075401152666789026648, 8.847076750739160820149839162568, 10.01965024261249855486569374232, 11.11020042829072547536979709980, 11.65664990667370187932815439230

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