Properties

Label 2-300-15.14-c2-0-7
Degree $2$
Conductor $300$
Sign $0.447 + 0.894i$
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  + 3i·3-s − 13i·7-s − 9·9-s − 23i·13-s − 11·19-s + 39·21-s − 27i·27-s + 59·31-s + 26i·37-s + 69·39-s − 83i·43-s − 120·49-s − 33i·57-s − 121·61-s + 117i·63-s + ⋯
L(s)  = 1  + i·3-s − 1.85i·7-s − 9-s − 1.76i·13-s − 0.578·19-s + 1.85·21-s i·27-s + 1.90·31-s + 0.702i·37-s + 1.76·39-s − 1.93i·43-s − 2.44·49-s − 0.578i·57-s − 1.98·61-s + 1.85i·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(8.17440\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1),\ 0.447 + 0.894i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.05048 - 0.649234i\)
\(L(\frac12)\) \(\approx\) \(1.05048 - 0.649234i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - 3iT \)
5 \( 1 \)
good7 \( 1 + 13iT - 49T^{2} \)
11 \( 1 - 121T^{2} \)
13 \( 1 + 23iT - 169T^{2} \)
17 \( 1 + 289T^{2} \)
19 \( 1 + 11T + 361T^{2} \)
23 \( 1 + 529T^{2} \)
29 \( 1 - 841T^{2} \)
31 \( 1 - 59T + 961T^{2} \)
37 \( 1 - 26iT - 1.36e3T^{2} \)
41 \( 1 - 1.68e3T^{2} \)
43 \( 1 + 83iT - 1.84e3T^{2} \)
47 \( 1 + 2.20e3T^{2} \)
53 \( 1 + 2.80e3T^{2} \)
59 \( 1 - 3.48e3T^{2} \)
61 \( 1 + 121T + 3.72e3T^{2} \)
67 \( 1 + 13iT - 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 - 46iT - 5.32e3T^{2} \)
79 \( 1 - 142T + 6.24e3T^{2} \)
83 \( 1 + 6.88e3T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 - 167iT - 9.40e3T^{2} \)
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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

−10.90882906484152064632781482593, −10.43376664125113216007951691107, −9.854263914352636454474409972376, −8.437165553783129398806432879654, −7.64802806328796646043794127308, −6.37137258114001689092462252030, −5.05874655275264717236550106787, −4.09292806883798939428159789023, −3.09567823811564449111867640423, −0.59979971103601221384346745151, 1.80714191230484323480230343056, 2.76878541219959199789426933710, 4.71065036191316452858260308615, 6.04468014680664510962976631743, 6.56719051155626428363894701120, 7.953258683561038491650617967762, 8.815381953638103562640972287861, 9.454965557907483102144459302987, 11.14313299260001934214916051504, 11.91893341119461219277812358815

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