Properties

Label 2-300-25.16-c1-0-2
Degree $2$
Conductor $300$
Sign $0.756 - 0.653i$
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 + (2.18 + 0.464i)5-s + 0.547·7-s + (−0.809 + 0.587i)9-s + (1.08 + 0.786i)11-s + (−0.244 + 0.177i)13-s + (0.233 + 2.22i)15-s + (1.24 − 3.84i)17-s + (−1.74 + 5.35i)19-s + (0.169 + 0.520i)21-s + (−0.198 − 0.144i)23-s + (4.56 + 2.03i)25-s + (−0.809 − 0.587i)27-s + (−0.423 − 1.30i)29-s + (−1.09 + 3.36i)31-s + ⋯
L(s)  = 1  + (0.178 + 0.549i)3-s + (0.978 + 0.207i)5-s + 0.206·7-s + (−0.269 + 0.195i)9-s + (0.326 + 0.237i)11-s + (−0.0677 + 0.0492i)13-s + (0.0603 + 0.574i)15-s + (0.302 − 0.932i)17-s + (−0.399 + 1.22i)19-s + (0.0369 + 0.113i)21-s + (−0.0413 − 0.0300i)23-s + (0.913 + 0.406i)25-s + (−0.155 − 0.113i)27-s + (−0.0785 − 0.241i)29-s + (−0.196 + 0.604i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 - 0.653i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.756 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.756 - 0.653i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.756 - 0.653i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49922 + 0.557555i\)
\(L(\frac12)\) \(\approx\) \(1.49922 + 0.557555i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.18 - 0.464i)T \)
good7 \( 1 - 0.547T + 7T^{2} \)
11 \( 1 + (-1.08 - 0.786i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.244 - 0.177i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.24 + 3.84i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.74 - 5.35i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (0.198 + 0.144i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (0.423 + 1.30i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.09 - 3.36i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.76 + 1.28i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-7.93 + 5.76i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 8.35T + 43T^{2} \)
47 \( 1 + (3.23 + 9.96i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (2.37 + 7.31i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (3.35 - 2.44i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (1.67 + 1.22i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-2.62 + 8.07i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (2.83 + 8.73i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (8.86 + 6.43i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-4.69 - 14.4i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (2.05 - 6.32i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (4.02 + 2.92i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-1.25 - 3.86i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.78297933299446024375426078338, −10.71928704886328491625821808530, −9.919015145471305123142232782293, −9.251500461885395594745291754053, −8.151534973228845726183433944582, −6.90129068606091381216781606330, −5.79657245122417011306687116138, −4.80952024617629072322721104249, −3.40496730084144707608362494979, −1.94752981332501780926111968456, 1.46395762947731839080065056928, 2.83650825450134602968918587389, 4.55211796492519213775239744477, 5.84763812531386659309057449303, 6.60278081244208892826074260969, 7.85004328182784129305609876963, 8.828367884063125816313934794884, 9.632588879550040190313735156663, 10.73811668212997914992856347722, 11.64168682083917800251936700443

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