Properties

Label 2-300-25.19-c1-0-5
Degree $2$
Conductor $300$
Sign $-0.628 + 0.778i$
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 + (−0.900 − 2.04i)5-s + 0.957i·7-s + (−0.309 + 0.951i)9-s + (−1.67 − 5.15i)11-s + (−1.92 − 0.625i)13-s + (−1.12 + 1.93i)15-s + (0.377 − 0.520i)17-s + (−4.07 − 2.96i)19-s + (0.774 − 0.562i)21-s + (−3.34 + 1.08i)23-s + (−3.37 + 3.68i)25-s + (0.951 − 0.309i)27-s + (8.20 − 5.96i)29-s + (−2.98 − 2.16i)31-s + ⋯
L(s)  = 1  + (−0.339 − 0.467i)3-s + (−0.402 − 0.915i)5-s + 0.361i·7-s + (−0.103 + 0.317i)9-s + (−0.504 − 1.55i)11-s + (−0.533 − 0.173i)13-s + (−0.290 + 0.498i)15-s + (0.0916 − 0.126i)17-s + (−0.935 − 0.679i)19-s + (0.169 − 0.122i)21-s + (−0.697 + 0.226i)23-s + (−0.675 + 0.737i)25-s + (0.183 − 0.0594i)27-s + (1.52 − 1.10i)29-s + (−0.536 − 0.389i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.628 + 0.778i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.628 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.628 + 0.778i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ -0.628 + 0.778i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.338879 - 0.709145i\)
\(L(\frac12)\) \(\approx\) \(0.338879 - 0.709145i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.900 + 2.04i)T \)
good7 \( 1 - 0.957iT - 7T^{2} \)
11 \( 1 + (1.67 + 5.15i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (1.92 + 0.625i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.377 + 0.520i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (4.07 + 2.96i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (3.34 - 1.08i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (-8.20 + 5.96i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.98 + 2.16i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-10.7 - 3.49i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.08 + 3.35i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 0.766iT - 43T^{2} \)
47 \( 1 + (-2.90 - 3.99i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (3.49 + 4.81i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.45 - 4.48i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.34 - 4.13i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (5.59 - 7.70i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (-9.66 + 7.02i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-5.16 + 1.67i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-9.58 + 6.96i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (0.819 - 1.12i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + (-0.527 - 1.62i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-8.57 - 11.8i)T + (-29.9 + 92.2i)T^{2} \)
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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

−11.57198191418209031309997430218, −10.67721135312075201056211933761, −9.379285565540139622501540319959, −8.366014144461551696671790841490, −7.80478272607340599388118367294, −6.29786208380862812435706545304, −5.46417051336255264063580714667, −4.30393985439625756257078226574, −2.61483693419786015736300993323, −0.58968769592860991218066089144, 2.39034550201853076812092023027, 3.93063295569729801889838101977, 4.82535887387697480424173515653, 6.30720905155842263699639229031, 7.19520571699066818039703331949, 8.088982802163805373248708580852, 9.590571247977398543943858909091, 10.33741308484352694646561758620, 10.89304590375103446551312084428, 12.14202338231551341712992086148

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