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Properties

Label 2-300-4.3-c2-0-14
Degree $2$
Conductor $300$
Sign $0.911 - 0.410i$
Analytic cond. $8.17440$
Root an. cond. $2.85909$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.67 − 1.08i)2-s + 1.73i·3-s + (1.64 + 3.64i)4-s + (1.87 − 2.90i)6-s + 0.596i·7-s + (1.19 − 7.91i)8-s − 2.99·9-s − 9.27i·11-s + (−6.31 + 2.84i)12-s + 23.5·13-s + (0.647 − 1.00i)14-s + (−10.5 + 11.9i)16-s − 3.97·17-s + (5.03 + 3.25i)18-s + 7.04i·19-s + ⋯
L(s)  = 1  + (−0.839 − 0.542i)2-s + 0.577i·3-s + (0.410 + 0.911i)4-s + (0.313 − 0.484i)6-s + 0.0852i·7-s + (0.149 − 0.988i)8-s − 0.333·9-s − 0.843i·11-s + (−0.526 + 0.237i)12-s + 1.80·13-s + (0.0462 − 0.0715i)14-s + (−0.662 + 0.749i)16-s − 0.233·17-s + (0.279 + 0.180i)18-s + 0.370i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.410i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.911 - 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.911 - 0.410i$
Analytic conductor: \(8.17440\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1),\ 0.911 - 0.410i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.09313 + 0.235003i\)
\(L(\frac12)\) \(\approx\) \(1.09313 + 0.235003i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.67 + 1.08i)T \)
3 \( 1 - 1.73iT \)
5 \( 1 \)
good7 \( 1 - 0.596iT - 49T^{2} \)
11 \( 1 + 9.27iT - 121T^{2} \)
13 \( 1 - 23.5T + 169T^{2} \)
17 \( 1 + 3.97T + 289T^{2} \)
19 \( 1 - 7.04iT - 361T^{2} \)
23 \( 1 - 32.0iT - 529T^{2} \)
29 \( 1 - 35.6T + 841T^{2} \)
31 \( 1 - 59.2iT - 961T^{2} \)
37 \( 1 - 5.38T + 1.36e3T^{2} \)
41 \( 1 - 40.0T + 1.68e3T^{2} \)
43 \( 1 - 36.1iT - 1.84e3T^{2} \)
47 \( 1 + 74.0iT - 2.20e3T^{2} \)
53 \( 1 - 2.55T + 2.80e3T^{2} \)
59 \( 1 + 36.4iT - 3.48e3T^{2} \)
61 \( 1 + 8.73T + 3.72e3T^{2} \)
67 \( 1 - 69.7iT - 4.48e3T^{2} \)
71 \( 1 - 59.2iT - 5.04e3T^{2} \)
73 \( 1 - 83.0T + 5.32e3T^{2} \)
79 \( 1 + 65.8iT - 6.24e3T^{2} \)
83 \( 1 - 129. iT - 6.88e3T^{2} \)
89 \( 1 + 130.T + 7.92e3T^{2} \)
97 \( 1 + 93.1T + 9.40e3T^{2} \)
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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

−11.22312258021516470393721538782, −10.74783728724734514756508771453, −9.744114911976376408380268826247, −8.702071781740209745340796758983, −8.278598804619897660433008973326, −6.79491782245011063342031901374, −5.66845549391618659142764005430, −3.95728969234595751249929480031, −3.08939236476823506431859599245, −1.22106835065571447854603991767, 0.890583935153528348516972048149, 2.37189247350972248843127245704, 4.40801758241639367304943323333, 5.95515042603119864366311524632, 6.60728068892630543544489089744, 7.67757573425561208386019790995, 8.511988999850259525934865990323, 9.349299852081076941629930989537, 10.53560500764075826075560057350, 11.19942034148470185924894078787

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