Properties

Label 2-300-12.11-c1-0-14
Degree $2$
Conductor $300$
Sign $0.791 + 0.611i$
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  + (−1.17 − 0.780i)2-s + (1.51 − 0.848i)3-s + (0.780 + 1.84i)4-s + (−2.44 − 0.179i)6-s + 3.02i·7-s + (0.516 − 2.78i)8-s + (1.56 − 2.56i)9-s + 1.32·11-s + (2.74 + 2.11i)12-s + 5.12·13-s + (2.35 − 3.56i)14-s + (−2.78 + 2.87i)16-s − 2i·17-s + (−3.84 + 1.80i)18-s + 1.32i·19-s + ⋯
L(s)  = 1  + (−0.833 − 0.552i)2-s + (0.871 − 0.489i)3-s + (0.390 + 0.920i)4-s + (−0.997 − 0.0731i)6-s + 1.14i·7-s + (0.182 − 0.983i)8-s + (0.520 − 0.853i)9-s + 0.399·11-s + (0.791 + 0.611i)12-s + 1.42·13-s + (0.630 − 0.951i)14-s + (−0.695 + 0.718i)16-s − 0.485i·17-s + (−0.905 + 0.424i)18-s + 0.303i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.791 + 0.611i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.791 + 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.791 + 0.611i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.791 + 0.611i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17873 - 0.402476i\)
\(L(\frac12)\) \(\approx\) \(1.17873 - 0.402476i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.17 + 0.780i)T \)
3 \( 1 + (-1.51 + 0.848i)T \)
5 \( 1 \)
good7 \( 1 - 3.02iT - 7T^{2} \)
11 \( 1 - 1.32T + 11T^{2} \)
13 \( 1 - 5.12T + 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 - 1.32iT - 19T^{2} \)
23 \( 1 + 0.371T + 23T^{2} \)
29 \( 1 - 3.12iT - 29T^{2} \)
31 \( 1 + 4.71iT - 31T^{2} \)
37 \( 1 + 5.12T + 37T^{2} \)
41 \( 1 + 1.12iT - 41T^{2} \)
43 \( 1 + 7.73iT - 43T^{2} \)
47 \( 1 + 3.02T + 47T^{2} \)
53 \( 1 - 12.2iT - 53T^{2} \)
59 \( 1 + 14.1T + 59T^{2} \)
61 \( 1 - 3.12T + 61T^{2} \)
67 \( 1 - 4.34iT - 67T^{2} \)
71 \( 1 - 3.39T + 71T^{2} \)
73 \( 1 + 8.24T + 73T^{2} \)
79 \( 1 - 8.10iT - 79T^{2} \)
83 \( 1 + 15.1T + 83T^{2} \)
89 \( 1 - 10.2iT - 89T^{2} \)
97 \( 1 - 6T + 97T^{2} \)
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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

−11.74763724835736426913013200276, −10.66237697499104107972629122938, −9.425202070938424417768502399993, −8.852496794027586327674932395755, −8.182948738038439037982727865948, −7.05608149256728945203329408191, −5.97273721688449261049088032353, −3.88413711283470981255955176605, −2.79038990651337344931056864529, −1.53736606286210913618572713178, 1.50222307256617399489668037956, 3.45823220178323003835659713269, 4.64225323437421005847002337631, 6.21281630154379773597355293168, 7.21950962792513002388126894916, 8.185214050457426792952148419244, 8.864689523324308638447612517608, 9.888195531669791162901254543061, 10.60099224243916951649087567544, 11.34031506185000029683653353792

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