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Properties

Label 2-2400-60.59-c1-0-61
Degree $2$
Conductor $2400$
Sign $-0.151 + 0.988i$
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  + (1.70 − 0.292i)3-s − 3.41·7-s + (2.82 − i)9-s + 2.82·11-s − 2i·13-s − 7.65·17-s + 2.82i·19-s + (−5.82 + i)21-s − 7.41i·23-s + (4.53 − 2.53i)27-s − 8i·29-s − 5.65i·31-s + (4.82 − 0.828i)33-s + 0.343i·37-s + (−0.585 − 3.41i)39-s + ⋯
L(s)  = 1  + (0.985 − 0.169i)3-s − 1.29·7-s + (0.942 − 0.333i)9-s + 0.852·11-s − 0.554i·13-s − 1.85·17-s + 0.648i·19-s + (−1.27 + 0.218i)21-s − 1.54i·23-s + (0.872 − 0.487i)27-s − 1.48i·29-s − 1.01i·31-s + (0.840 − 0.144i)33-s + 0.0564i·37-s + (−0.0938 − 0.546i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.151 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2400\)    =    \(2^{5} \cdot 3 \cdot 5^{2}\)
Sign: $-0.151 + 0.988i$
Analytic conductor: \(19.1640\)
Root analytic conductor: \(4.37768\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2400} (2399, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2400,\ (\ :1/2),\ -0.151 + 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.742700498\)
\(L(\frac12)\) \(\approx\) \(1.742700498\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.70 + 0.292i)T \)
5 \( 1 \)
good7 \( 1 + 3.41T + 7T^{2} \)
11 \( 1 - 2.82T + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + 7.65T + 17T^{2} \)
19 \( 1 - 2.82iT - 19T^{2} \)
23 \( 1 + 7.41iT - 23T^{2} \)
29 \( 1 + 8iT - 29T^{2} \)
31 \( 1 + 5.65iT - 31T^{2} \)
37 \( 1 - 0.343iT - 37T^{2} \)
41 \( 1 + 2iT - 41T^{2} \)
43 \( 1 - 7.89T + 43T^{2} \)
47 \( 1 + 6.24iT - 47T^{2} \)
53 \( 1 + 3.65T + 53T^{2} \)
59 \( 1 - 1.65T + 59T^{2} \)
61 \( 1 - 5.65T + 61T^{2} \)
67 \( 1 - 1.07T + 67T^{2} \)
71 \( 1 + 1.17T + 71T^{2} \)
73 \( 1 - 15.6iT - 73T^{2} \)
79 \( 1 - 4.48iT - 79T^{2} \)
83 \( 1 + 5.07iT - 83T^{2} \)
89 \( 1 + 7.31iT - 89T^{2} \)
97 \( 1 + 18.9iT - 97T^{2} \)
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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

−8.724883395859655719714803863985, −8.184463292276824541642759071099, −7.10648040450643913356269732932, −6.55669595309024432048309908971, −5.91901269074076452941616561502, −4.31588032579121051688519370677, −3.93885803989304501445664910713, −2.81057384900488330645414989652, −2.14218418945871467103825991766, −0.49964425493072490125387421834, 1.48095251425234716265534826934, 2.60655695855997280987215329255, 3.44749087836661785750565132887, 4.13466273104950444861674502415, 5.06214755876157013257760103599, 6.46480980624158757859443512782, 6.77548446262113259592458978357, 7.57070339968828834644997053076, 8.745394414240392457038587300619, 9.259458664693870515968605546480

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