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Properties

Label 2-300-4.3-c4-0-27
Degree $2$
Conductor $300$
Sign $0.280 - 0.959i$
Analytic cond. $31.0109$
Root an. cond. $5.56875$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.95 + 0.566i)2-s − 5.19i·3-s + (15.3 + 4.48i)4-s + (2.94 − 20.5i)6-s + 24.1i·7-s + (58.2 + 26.4i)8-s − 27·9-s + 227. i·11-s + (23.2 − 79.8i)12-s − 285.·13-s + (−13.6 + 95.5i)14-s + (215. + 137. i)16-s + 301.·17-s + (−106. − 15.2i)18-s + 674. i·19-s + ⋯
L(s)  = 1  + (0.989 + 0.141i)2-s − 0.577i·3-s + (0.959 + 0.280i)4-s + (0.0816 − 0.571i)6-s + 0.492i·7-s + (0.910 + 0.413i)8-s − 0.333·9-s + 1.87i·11-s + (0.161 − 0.554i)12-s − 1.68·13-s + (−0.0697 + 0.487i)14-s + (0.843 + 0.537i)16-s + 1.04·17-s + (−0.329 − 0.0471i)18-s + 1.86i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.280 - 0.959i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.280 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.280 - 0.959i$
Analytic conductor: \(31.0109\)
Root analytic conductor: \(5.56875\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :2),\ 0.280 - 0.959i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(3.277625528\)
\(L(\frac12)\) \(\approx\) \(3.277625528\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.95 - 0.566i)T \)
3 \( 1 + 5.19iT \)
5 \( 1 \)
good7 \( 1 - 24.1iT - 2.40e3T^{2} \)
11 \( 1 - 227. iT - 1.46e4T^{2} \)
13 \( 1 + 285.T + 2.85e4T^{2} \)
17 \( 1 - 301.T + 8.35e4T^{2} \)
19 \( 1 - 674. iT - 1.30e5T^{2} \)
23 \( 1 + 459. iT - 2.79e5T^{2} \)
29 \( 1 - 146.T + 7.07e5T^{2} \)
31 \( 1 + 702. iT - 9.23e5T^{2} \)
37 \( 1 - 100.T + 1.87e6T^{2} \)
41 \( 1 - 1.10e3T + 2.82e6T^{2} \)
43 \( 1 - 811. iT - 3.41e6T^{2} \)
47 \( 1 - 1.19e3iT - 4.87e6T^{2} \)
53 \( 1 + 2.29e3T + 7.89e6T^{2} \)
59 \( 1 - 3.14e3iT - 1.21e7T^{2} \)
61 \( 1 + 2.02e3T + 1.38e7T^{2} \)
67 \( 1 - 2.69e3iT - 2.01e7T^{2} \)
71 \( 1 + 8.36e3iT - 2.54e7T^{2} \)
73 \( 1 - 6.72e3T + 2.83e7T^{2} \)
79 \( 1 + 2.83e3iT - 3.89e7T^{2} \)
83 \( 1 - 4.62e3iT - 4.74e7T^{2} \)
89 \( 1 + 245.T + 6.27e7T^{2} \)
97 \( 1 + 1.57e3T + 8.85e7T^{2} \)
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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

−11.92175152275129444707699241568, −10.36984351313324362184206711572, −9.626754971835200543834659541301, −7.84789027592537433553845347387, −7.43775360829471773380387126428, −6.28503257683714278004507366400, −5.24986209499579329530181709988, −4.27608367356502856287846665411, −2.67512942768698936636435268593, −1.77769162527465717728223018604, 0.69105831825206306782304215804, 2.71101129406711575714753798798, 3.55877842217803042997519670037, 4.85282665916423342710137342174, 5.56856392019337908937523339305, 6.83452583108822863632901073001, 7.82277355683658885583429002682, 9.209487163937724781282467698788, 10.22928215338338142544081977733, 11.07338180159491993517959625451

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