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Properties

Label 2-4140-15.2-c1-0-41
Degree $2$
Conductor $4140$
Sign $-0.777 - 0.628i$
Analytic cond. $33.0580$
Root an. cond. $5.74961$
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  + (−2.20 + 0.371i)5-s + (−3.41 − 3.41i)7-s − 4.26i·11-s + (−1.36 + 1.36i)13-s + (3.76 − 3.76i)17-s − 3.34i·19-s + (−0.707 − 0.707i)23-s + (4.72 − 1.63i)25-s − 4.28·29-s + 1.17·31-s + (8.80 + 6.26i)35-s + (−1.12 − 1.12i)37-s − 10.1i·41-s + (8.84 − 8.84i)43-s + (−4.85 + 4.85i)47-s + ⋯
L(s)  = 1  + (−0.986 + 0.166i)5-s + (−1.29 − 1.29i)7-s − 1.28i·11-s + (−0.379 + 0.379i)13-s + (0.912 − 0.912i)17-s − 0.768i·19-s + (−0.147 − 0.147i)23-s + (0.944 − 0.327i)25-s − 0.796·29-s + 0.211·31-s + (1.48 + 1.05i)35-s + (−0.184 − 0.184i)37-s − 1.58i·41-s + (1.34 − 1.34i)43-s + (−0.707 + 0.707i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.777 - 0.628i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.777 - 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $-0.777 - 0.628i$
Analytic conductor: \(33.0580\)
Root analytic conductor: \(5.74961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4140} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4140,\ (\ :1/2),\ -0.777 - 0.628i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4399894204\)
\(L(\frac12)\) \(\approx\) \(0.4399894204\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (2.20 - 0.371i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + (3.41 + 3.41i)T + 7iT^{2} \)
11 \( 1 + 4.26iT - 11T^{2} \)
13 \( 1 + (1.36 - 1.36i)T - 13iT^{2} \)
17 \( 1 + (-3.76 + 3.76i)T - 17iT^{2} \)
19 \( 1 + 3.34iT - 19T^{2} \)
29 \( 1 + 4.28T + 29T^{2} \)
31 \( 1 - 1.17T + 31T^{2} \)
37 \( 1 + (1.12 + 1.12i)T + 37iT^{2} \)
41 \( 1 + 10.1iT - 41T^{2} \)
43 \( 1 + (-8.84 + 8.84i)T - 43iT^{2} \)
47 \( 1 + (4.85 - 4.85i)T - 47iT^{2} \)
53 \( 1 + (7.95 + 7.95i)T + 53iT^{2} \)
59 \( 1 - 4.65T + 59T^{2} \)
61 \( 1 + 12.2T + 61T^{2} \)
67 \( 1 + (-9.27 - 9.27i)T + 67iT^{2} \)
71 \( 1 + 7.18iT - 71T^{2} \)
73 \( 1 + (0.225 - 0.225i)T - 73iT^{2} \)
79 \( 1 - 10.6iT - 79T^{2} \)
83 \( 1 + (7.70 + 7.70i)T + 83iT^{2} \)
89 \( 1 - 3.97T + 89T^{2} \)
97 \( 1 + (0.874 + 0.874i)T + 97iT^{2} \)
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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

−7.74100887700097540157238609320, −7.25764391718814494378020167489, −6.72385573329425729718315488053, −5.85357990120747012297374952311, −4.86740391666755198322027559049, −3.86099497971841778055661221082, −3.48578573764201640007971771964, −2.68394543304185662092924405453, −0.806366139189215920117927018668, −0.17486851892450682970338187744, 1.57232071053046286426702179199, 2.75778731611385341292672851708, 3.40399953358352282890041146655, 4.27922165854476059802684188737, 5.17029509640890321818557183297, 5.98536319224015418869968203873, 6.58015054152892026754061569183, 7.62896161130312167924017779022, 7.959388502010042861722966054847, 8.881115651305066987884001947943

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