Properties

Label 2-300-25.4-c1-0-3
Degree $2$
Conductor $300$
Sign $0.611 + 0.791i$
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  + (0.587 − 0.809i)3-s + (−0.921 + 2.03i)5-s − 4.41i·7-s + (−0.309 − 0.951i)9-s + (1.37 − 4.23i)11-s + (5.46 − 1.77i)13-s + (1.10 + 1.94i)15-s + (3.86 + 5.31i)17-s + (−2.25 + 1.63i)19-s + (−3.57 − 2.59i)21-s + (1.25 + 0.406i)23-s + (−3.30 − 3.75i)25-s + (−0.951 − 0.309i)27-s + (−3.91 − 2.84i)29-s + (0.159 − 0.115i)31-s + ⋯
L(s)  = 1  + (0.339 − 0.467i)3-s + (−0.412 + 0.911i)5-s − 1.66i·7-s + (−0.103 − 0.317i)9-s + (0.414 − 1.27i)11-s + (1.51 − 0.492i)13-s + (0.285 + 0.501i)15-s + (0.936 + 1.28i)17-s + (−0.516 + 0.375i)19-s + (−0.779 − 0.566i)21-s + (0.260 + 0.0847i)23-s + (−0.660 − 0.751i)25-s + (−0.183 − 0.0594i)27-s + (−0.727 − 0.528i)29-s + (0.0286 − 0.0208i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.611 + 0.791i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.611 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.611 + 0.791i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.611 + 0.791i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25545 - 0.616131i\)
\(L(\frac12)\) \(\approx\) \(1.25545 - 0.616131i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.587 + 0.809i)T \)
5 \( 1 + (0.921 - 2.03i)T \)
good7 \( 1 + 4.41iT - 7T^{2} \)
11 \( 1 + (-1.37 + 4.23i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-5.46 + 1.77i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-3.86 - 5.31i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.25 - 1.63i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-1.25 - 0.406i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (3.91 + 2.84i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.159 + 0.115i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (7.80 - 2.53i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.42 - 7.46i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 0.412iT - 43T^{2} \)
47 \( 1 + (4.58 - 6.30i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.185 - 0.255i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.778 + 2.39i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.88 + 8.87i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-7.02 - 9.66i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (0.411 + 0.299i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-14.9 - 4.87i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (2.77 + 2.01i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-3.15 - 4.34i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (3.50 - 10.7i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-2.98 + 4.10i)T + (-29.9 - 92.2i)T^{2} \)
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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

−11.22521209534758247559650798463, −10.88709234446681362750708813731, −9.969216303095728876576724462478, −8.318906949306929758536034258358, −7.916588209201452348026152103219, −6.69558353024039498847493077838, −6.02359646940694062017406652028, −3.76599172277305923753824093051, −3.49084372258075948586677442346, −1.16750881501620176286126686030, 1.94216045530349824829097192932, 3.56621178305207407891381430233, 4.82256227984775760770941331370, 5.64078486706985875157705859492, 7.10537490419863691112504449990, 8.454448130697331600734617262789, 9.012445609098090481280153236476, 9.624439113121098216305774480145, 11.11734634204796405228120973926, 12.02033634863585524965296273453

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