Properties

Label 2-300-4.3-c4-0-40
Degree $2$
Conductor $300$
Sign $0.356 - 0.934i$
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  + (−0.725 + 3.93i)2-s + 5.19i·3-s + (−14.9 − 5.70i)4-s + (−20.4 − 3.76i)6-s + 36.7i·7-s + (33.2 − 54.6i)8-s − 27·9-s − 156. i·11-s + (29.6 − 77.6i)12-s + 62.1·13-s + (−144. − 26.6i)14-s + (190. + 170. i)16-s + 306.·17-s + (19.5 − 106. i)18-s − 242. i·19-s + ⋯
L(s)  = 1  + (−0.181 + 0.983i)2-s + 0.577i·3-s + (−0.934 − 0.356i)4-s + (−0.567 − 0.104i)6-s + 0.749i·7-s + (0.520 − 0.854i)8-s − 0.333·9-s − 1.29i·11-s + (0.205 − 0.539i)12-s + 0.367·13-s + (−0.737 − 0.135i)14-s + (0.745 + 0.666i)16-s + 1.06·17-s + (0.0604 − 0.327i)18-s − 0.672i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.356 - 0.934i$
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: \((2,\ 300,\ (\ :2),\ 0.356 - 0.934i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.560180832\)
\(L(\frac12)\) \(\approx\) \(1.560180832\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.725 - 3.93i)T \)
3 \( 1 - 5.19iT \)
5 \( 1 \)
good7 \( 1 - 36.7iT - 2.40e3T^{2} \)
11 \( 1 + 156. iT - 1.46e4T^{2} \)
13 \( 1 - 62.1T + 2.85e4T^{2} \)
17 \( 1 - 306.T + 8.35e4T^{2} \)
19 \( 1 + 242. iT - 1.30e5T^{2} \)
23 \( 1 + 571. iT - 2.79e5T^{2} \)
29 \( 1 + 266.T + 7.07e5T^{2} \)
31 \( 1 - 759. iT - 9.23e5T^{2} \)
37 \( 1 - 1.21e3T + 1.87e6T^{2} \)
41 \( 1 - 2.67e3T + 2.82e6T^{2} \)
43 \( 1 + 2.63e3iT - 3.41e6T^{2} \)
47 \( 1 - 2.37e3iT - 4.87e6T^{2} \)
53 \( 1 - 1.39e3T + 7.89e6T^{2} \)
59 \( 1 - 2.90e3iT - 1.21e7T^{2} \)
61 \( 1 + 4.78e3T + 1.38e7T^{2} \)
67 \( 1 - 4.65e3iT - 2.01e7T^{2} \)
71 \( 1 - 6.76e3iT - 2.54e7T^{2} \)
73 \( 1 + 6.58e3T + 2.83e7T^{2} \)
79 \( 1 + 8.54e3iT - 3.89e7T^{2} \)
83 \( 1 + 2.17e3iT - 4.74e7T^{2} \)
89 \( 1 - 6.63e3T + 6.27e7T^{2} \)
97 \( 1 - 1.57e4T + 8.85e7T^{2} \)
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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

−11.06455231948846162635901207921, −10.20220881325289272371382801899, −9.003797648762489066786476432356, −8.637772967948839111284317547288, −7.49023242552430460519883214673, −6.09512054358514332280331629312, −5.57020034799083404814935955148, −4.32522795580335245858069662311, −3.01381367589956022697706474076, −0.73065859758307201244181656067, 0.923739915938365217738454587242, 1.97221866144395689209771681054, 3.43950858531853654181416999004, 4.50131873464180780986927589458, 5.85682530310940256046317446581, 7.44721002616781350557105598461, 7.88296279836436712465194158424, 9.353844294368149969045408762732, 9.993530775988484544577147885399, 10.97867158337096083930852441085

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