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Properties

Label 2-2400-120.59-c1-0-32
Degree $2$
Conductor $2400$
Sign $0.949 - 0.314i$
Analytic cond. $19.1640$
Root an. cond. $4.37768$
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  + (−1.71 + 0.242i)3-s + 3.08·7-s + (2.88 − 0.831i)9-s + 2.54i·11-s + 5.06·13-s + 0.214·17-s + 2.60·19-s + (−5.29 + 0.749i)21-s − 4.47i·23-s + (−4.74 + 2.12i)27-s + 7.86·29-s + 4.58i·31-s + (−0.617 − 4.36i)33-s − 7.67·37-s + (−8.68 + 1.22i)39-s + ⋯
L(s)  = 1  + (−0.990 + 0.139i)3-s + 1.16·7-s + (0.960 − 0.277i)9-s + 0.767i·11-s + 1.40·13-s + 0.0519·17-s + 0.598·19-s + (−1.15 + 0.163i)21-s − 0.933i·23-s + (−0.912 + 0.408i)27-s + 1.46·29-s + 0.823i·31-s + (−0.107 − 0.760i)33-s − 1.26·37-s + (−1.39 + 0.196i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.314i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.949 - 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2400\)    =    \(2^{5} \cdot 3 \cdot 5^{2}\)
Sign: $0.949 - 0.314i$
Analytic conductor: \(19.1640\)
Root analytic conductor: \(4.37768\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2400} (1199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2400,\ (\ :1/2),\ 0.949 - 0.314i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.681154407\)
\(L(\frac12)\) \(\approx\) \(1.681154407\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.71 - 0.242i)T \)
5 \( 1 \)
good7 \( 1 - 3.08T + 7T^{2} \)
11 \( 1 - 2.54iT - 11T^{2} \)
13 \( 1 - 5.06T + 13T^{2} \)
17 \( 1 - 0.214T + 17T^{2} \)
19 \( 1 - 2.60T + 19T^{2} \)
23 \( 1 + 4.47iT - 23T^{2} \)
29 \( 1 - 7.86T + 29T^{2} \)
31 \( 1 - 4.58iT - 31T^{2} \)
37 \( 1 + 7.67T + 37T^{2} \)
41 \( 1 - 9.26iT - 41T^{2} \)
43 \( 1 + 11.4iT - 43T^{2} \)
47 \( 1 + 10.5iT - 47T^{2} \)
53 \( 1 - 9.51iT - 53T^{2} \)
59 \( 1 - 0.428iT - 59T^{2} \)
61 \( 1 + 1.11iT - 61T^{2} \)
67 \( 1 - 2.35iT - 67T^{2} \)
71 \( 1 + 6.12T + 71T^{2} \)
73 \( 1 + 12.0iT - 73T^{2} \)
79 \( 1 - 11.6iT - 79T^{2} \)
83 \( 1 - 2.29T + 83T^{2} \)
89 \( 1 + 12.4iT - 89T^{2} \)
97 \( 1 - 8.04iT - 97T^{2} \)
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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

−8.842253599988329848141220761247, −8.335320854111865173699801849020, −7.31578280131185085379244115198, −6.66526507325241735761584458164, −5.82487347808285846792108283871, −4.95624878421110219831102232160, −4.48895587238277888286601269762, −3.42996918038408259260312384111, −1.87944964263194770335189572183, −0.996535011532860407003201481051, 0.917321976110004157474147405869, 1.70459494217648759246507309254, 3.27208451284731473506940325666, 4.25607708417770555362138311032, 5.09595048067096958507684757074, 5.79202203416193630861216813950, 6.42176428773582321693157230897, 7.42135804054283443905140280782, 8.115834821267755604969787708616, 8.773364180696285621133394697380

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