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Properties

Label 2-300-5.2-c2-0-4
Degree $2$
Conductor $300$
Sign $0.229 + 0.973i$
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  + (1.22 − 1.22i)3-s + (−2.44 − 2.44i)7-s − 2.99i·9-s + 6·11-s + (12.2 − 12.2i)13-s + (−14.6 − 14.6i)17-s − 10i·19-s − 5.99·21-s + (29.3 − 29.3i)23-s + (−3.67 − 3.67i)27-s + 48i·29-s − 26·31-s + (7.34 − 7.34i)33-s + (−31.8 − 31.8i)37-s − 29.9i·39-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (−0.349 − 0.349i)7-s − 0.333i·9-s + 0.545·11-s + (0.942 − 0.942i)13-s + (−0.864 − 0.864i)17-s − 0.526i·19-s − 0.285·21-s + (1.27 − 1.27i)23-s + (−0.136 − 0.136i)27-s + 1.65i·29-s − 0.838·31-s + (0.222 − 0.222i)33-s + (−0.860 − 0.860i)37-s − 0.769i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.229 + 0.973i$
Analytic conductor: \(8.17440\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1),\ 0.229 + 0.973i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.37257 - 1.08628i\)
\(L(\frac12)\) \(\approx\) \(1.37257 - 1.08628i\)
\(L(2)\) 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 \( 1 + (-1.22 + 1.22i)T \)
5 \( 1 \)
good7 \( 1 + (2.44 + 2.44i)T + 49iT^{2} \)
11 \( 1 - 6T + 121T^{2} \)
13 \( 1 + (-12.2 + 12.2i)T - 169iT^{2} \)
17 \( 1 + (14.6 + 14.6i)T + 289iT^{2} \)
19 \( 1 + 10iT - 361T^{2} \)
23 \( 1 + (-29.3 + 29.3i)T - 529iT^{2} \)
29 \( 1 - 48iT - 841T^{2} \)
31 \( 1 + 26T + 961T^{2} \)
37 \( 1 + (31.8 + 31.8i)T + 1.36e3iT^{2} \)
41 \( 1 - 30T + 1.68e3T^{2} \)
43 \( 1 + (-29.3 + 29.3i)T - 1.84e3iT^{2} \)
47 \( 1 + (-14.6 - 14.6i)T + 2.20e3iT^{2} \)
53 \( 1 + (14.6 - 14.6i)T - 2.80e3iT^{2} \)
59 \( 1 - 78iT - 3.48e3T^{2} \)
61 \( 1 - 2T + 3.72e3T^{2} \)
67 \( 1 + (-63.6 - 63.6i)T + 4.48e3iT^{2} \)
71 \( 1 - 120T + 5.04e3T^{2} \)
73 \( 1 + (83.2 - 83.2i)T - 5.32e3iT^{2} \)
79 \( 1 + 74iT - 6.24e3T^{2} \)
83 \( 1 + (44.0 - 44.0i)T - 6.88e3iT^{2} \)
89 \( 1 - 150iT - 7.92e3T^{2} \)
97 \( 1 + (4.89 + 4.89i)T + 9.40e3iT^{2} \)
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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

−11.16357764241274939275287442941, −10.56450376971257587253596129056, −9.113649471891719120977312023371, −8.692556380368968603604533547914, −7.26396201705911755964380760743, −6.66975982072101805302522291877, −5.30175885257502700729050973247, −3.86058894147713472485762650239, −2.69214388207716596739459407057, −0.868602138132027718415880162113, 1.78845034754583931206330064212, 3.42030320860694633127824883749, 4.35087548020514904070469427966, 5.83453145277874776818972635963, 6.75243954753592383032685549299, 8.079404660715821071305160262429, 9.053721196811788048756463160260, 9.582145368337176421978781038027, 10.89800635934647474438992310690, 11.52691969320964188916900906915

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