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Properties

Label 2-3420-285.227-c0-0-7
Degree $2$
Conductor $3420$
Sign $-0.990 - 0.139i$
Analytic cond. $1.70680$
Root an. cond. $1.30644$
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  + (−0.258 − 0.965i)5-s + (−1.36 − 1.36i)7-s − 1.93i·11-s + (−0.707 + 0.707i)17-s + i·19-s + (−0.866 + 0.499i)25-s + (−0.965 + 1.67i)35-s + (−1.36 + 1.36i)43-s + (1.22 − 1.22i)47-s + 2.73i·49-s + (−1.86 + 0.499i)55-s + 61-s + (−0.366 + 0.366i)73-s + (−2.63 + 2.63i)77-s + (−1.41 − 1.41i)83-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)5-s + (−1.36 − 1.36i)7-s − 1.93i·11-s + (−0.707 + 0.707i)17-s + i·19-s + (−0.866 + 0.499i)25-s + (−0.965 + 1.67i)35-s + (−1.36 + 1.36i)43-s + (1.22 − 1.22i)47-s + 2.73i·49-s + (−1.86 + 0.499i)55-s + 61-s + (−0.366 + 0.366i)73-s + (−2.63 + 2.63i)77-s + (−1.41 − 1.41i)83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3420\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $-0.990 - 0.139i$
Analytic conductor: \(1.70680\)
Root analytic conductor: \(1.30644\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3420} (3077, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3420,\ (\ :0),\ -0.990 - 0.139i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5143631379\)
\(L(\frac12)\) \(\approx\) \(0.5143631379\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (0.258 + 0.965i)T \)
19 \( 1 - iT \)
good7 \( 1 + (1.36 + 1.36i)T + iT^{2} \)
11 \( 1 + 1.93iT - T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (1.36 - 1.36i)T - iT^{2} \)
47 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
89 \( 1 - T^{2} \)
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

−8.438370762152526111804821067853, −7.78053509994837768083992784185, −6.81694279092104228701933939525, −6.14803977278041848191696613181, −5.51430490042788972606513826618, −4.26439159600123168862638164766, −3.75205251474023322678618222852, −3.06751056731831642776558594651, −1.36349835383364741347430811139, −0.30332902341634808768270322153, 2.27127599172579758269123387846, 2.56640689119554088322715137960, 3.57835420325923909536292460833, 4.59010677291580774392309409535, 5.44278778393484696989368753979, 6.41031755584598911449972816316, 6.92115171061132835452670872379, 7.37379413713436097791848374805, 8.584298591017360517429933217694, 9.300310965079742376159156011805

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