Properties

Label 2-3120-3120.1403-c0-0-3
Degree $2$
Conductor $3120$
Sign $0.987 + 0.160i$
Analytic cond. $1.55708$
Root an. cond. $1.24783$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)2-s + i·3-s + 1.00i·4-s + (−0.707 − 0.707i)5-s + (0.707 − 0.707i)6-s + (0.707 − 0.707i)8-s − 9-s + 1.00i·10-s − 1.00·12-s + 13-s + (0.707 − 0.707i)15-s − 1.00·16-s + (0.707 + 0.707i)18-s + (0.707 − 0.707i)20-s + (0.707 + 0.707i)24-s + 1.00i·25-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + i·3-s + 1.00i·4-s + (−0.707 − 0.707i)5-s + (0.707 − 0.707i)6-s + (0.707 − 0.707i)8-s − 9-s + 1.00i·10-s − 1.00·12-s + 13-s + (0.707 − 0.707i)15-s − 1.00·16-s + (0.707 + 0.707i)18-s + (0.707 − 0.707i)20-s + (0.707 + 0.707i)24-s + 1.00i·25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3120\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.987 + 0.160i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3120} (1403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3120,\ (\ :0),\ 0.987 + 0.160i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7250815888\)
\(L(\frac12)\) \(\approx\) \(0.7250815888\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 - iT \)
5 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 - T \)
good7 \( 1 + iT^{2} \)
11 \( 1 - iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (-1 + i)T - iT^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - 1.41T + T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - 2T + T^{2} \)
83 \( 1 - 1.41iT - T^{2} \)
89 \( 1 + 1.41T + T^{2} \)
97 \( 1 + iT^{2} \)
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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

−8.959575649681308491365574519738, −8.358470653430911102815862757443, −7.78818486088298097495705782549, −6.74339155525202695618909673898, −5.60811050288213643409485907942, −4.74689988949350776438952378874, −3.87206190017888613490066255873, −3.51621297133632390362146193382, −2.27885732909406712972943687754, −0.838282418099658473660954116707, 0.870082586072284331922961752546, 2.07999575465429668693267552388, 3.14527855160455437024303906252, 4.25645840718611188655040479630, 5.46934026274358945522738515109, 6.18628122390421954542041919613, 6.79542471563071772365833722645, 7.38011407248040072540568691554, 8.116844839162447758444193624453, 8.531655447206990704210071528805

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