Properties

Label 2-3120-5.4-c1-0-43
Degree $2$
Conductor $3120$
Sign $-0.158 + 0.987i$
Analytic cond. $24.9133$
Root an. cond. $4.99132$
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  i·3-s + (−0.353 + 2.20i)5-s + 3.09i·7-s − 9-s − 5.51·11-s i·13-s + (2.20 + 0.353i)15-s − 6.21i·17-s + 3.09·21-s + 4.21i·23-s + (−4.74 − 1.56i)25-s + i·27-s + 1.70·29-s + 5.51i·33-s + (−6.83 − 1.09i)35-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−0.158 + 0.987i)5-s + 1.16i·7-s − 0.333·9-s − 1.66·11-s − 0.277i·13-s + (0.570 + 0.0913i)15-s − 1.50i·17-s + 0.675·21-s + 0.879i·23-s + (−0.949 − 0.312i)25-s + 0.192i·27-s + 0.317·29-s + 0.959i·33-s + (−1.15 − 0.184i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.158 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.158 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3120\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.158 + 0.987i$
Analytic conductor: \(24.9133\)
Root analytic conductor: \(4.99132\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3120} (1249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3120,\ (\ :1/2),\ -0.158 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6591299520\)
\(L(\frac12)\) \(\approx\) \(0.6591299520\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
5 \( 1 + (0.353 - 2.20i)T \)
13 \( 1 + iT \)
good7 \( 1 - 3.09iT - 7T^{2} \)
11 \( 1 + 5.51T + 11T^{2} \)
17 \( 1 + 6.21iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 4.21iT - 23T^{2} \)
29 \( 1 - 1.70T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 4.02iT - 37T^{2} \)
41 \( 1 - 0.198T + 41T^{2} \)
43 \( 1 + 5.70iT - 43T^{2} \)
47 \( 1 - 6.41iT - 47T^{2} \)
53 \( 1 + 4.02iT - 53T^{2} \)
59 \( 1 + 7.53T + 59T^{2} \)
61 \( 1 - 11.9T + 61T^{2} \)
67 \( 1 + 4.83iT - 67T^{2} \)
71 \( 1 + 7.98T + 71T^{2} \)
73 \( 1 + 9.31iT - 73T^{2} \)
79 \( 1 - 8.51T + 79T^{2} \)
83 \( 1 + 14.7iT - 83T^{2} \)
89 \( 1 - 9.40T + 89T^{2} \)
97 \( 1 - 14.3iT - 97T^{2} \)
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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

−8.328598920458842964556354326138, −7.58172926218097402152333061883, −7.21322042925981245471906843017, −6.15654892356302709900595257116, −5.53197201788059060745264676589, −4.87490188059973926959457421079, −3.32873240645577672483565968809, −2.71039585267873763470712031107, −2.10065829592837118655949822045, −0.22447652391590633429317048679, 1.03921646776115361471717209972, 2.36248764995748573307530239664, 3.58341198009767310795105480982, 4.30340885097817407273187894169, 4.89769666884773485199101022056, 5.68773367131032109904761694149, 6.62028335349042304410717046351, 7.66148298874686436508471564608, 8.188593629209589241085243951934, 8.738589910775866032484805010074

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