Properties

Label 2-300-25.4-c1-0-1
Degree $2$
Conductor $300$
Sign $0.725 - 0.687i$
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 + (0.892 + 2.05i)5-s + 4.13i·7-s + (−0.309 − 0.951i)9-s + (−1.16 + 3.58i)11-s + (0.664 − 0.215i)13-s + (2.18 + 0.482i)15-s + (−3.11 − 4.28i)17-s + (4.63 − 3.37i)19-s + (3.34 + 2.42i)21-s + (5.19 + 1.68i)23-s + (−3.40 + 3.66i)25-s + (−0.951 − 0.309i)27-s + (−5.68 − 4.12i)29-s + (8.16 − 5.93i)31-s + ⋯
L(s)  = 1  + (0.339 − 0.467i)3-s + (0.399 + 0.916i)5-s + 1.56i·7-s + (−0.103 − 0.317i)9-s + (−0.350 + 1.08i)11-s + (0.184 − 0.0598i)13-s + (0.563 + 0.124i)15-s + (−0.754 − 1.03i)17-s + (1.06 − 0.773i)19-s + (0.729 + 0.530i)21-s + (1.08 + 0.352i)23-s + (−0.681 + 0.732i)25-s + (−0.183 − 0.0594i)27-s + (−1.05 − 0.766i)29-s + (1.46 − 1.06i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.725 - 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.725 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.725 - 0.687i$
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: \((2,\ 300,\ (\ :1/2),\ 0.725 - 0.687i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37748 + 0.549149i\)
\(L(\frac12)\) \(\approx\) \(1.37748 + 0.549149i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.892 - 2.05i)T \)
good7 \( 1 - 4.13iT - 7T^{2} \)
11 \( 1 + (1.16 - 3.58i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.664 + 0.215i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (3.11 + 4.28i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-4.63 + 3.37i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-5.19 - 1.68i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (5.68 + 4.12i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-8.16 + 5.93i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-5.50 + 1.78i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.03 + 6.27i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 4.79iT - 43T^{2} \)
47 \( 1 + (5.68 - 7.82i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.99 - 2.74i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.230 + 0.708i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-3.64 + 11.2i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (2.81 + 3.88i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (-2.54 - 1.84i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-10.2 - 3.32i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (7.38 + 5.36i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (4.96 + 6.82i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (-1.04 + 3.22i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (6.13 - 8.44i)T + (-29.9 - 92.2i)T^{2} \)
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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

−11.72211376960276837251327227442, −11.20680718693125717245846688287, −9.548952442792454249781876522927, −9.360814555300916045504483722605, −7.891645717089069537635658725139, −7.02072318742235492384038808931, −6.02513680741255936708925659548, −4.92297976372415775440930799829, −2.91267675505473961953426885759, −2.24092502285280655608692705374, 1.19709850597878694792946333474, 3.34258410744677071103678213648, 4.37711248611742869223028912357, 5.45288998164262372941114547294, 6.75207276074516998305883954273, 8.061245709256135432254906790392, 8.699609200946904874459010603574, 9.862697422305283593978686451288, 10.55599526123296328780665222379, 11.41816327209445857421755566974

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