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Properties

Label 2-3420-3420.1139-c0-0-8
Degree $2$
Conductor $3420$
Sign $-0.766 + 0.642i$
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  + (−0.608 − 0.793i)2-s + (−0.130 − 0.991i)3-s + (−0.258 + 0.965i)4-s + (−0.866 − 0.5i)5-s + (−0.707 + 0.707i)6-s + (0.923 − 0.382i)8-s + (−0.965 + 0.258i)9-s + (0.130 + 0.991i)10-s + (0.965 + 1.67i)11-s + (0.991 + 0.130i)12-s + (0.608 − 1.05i)13-s + (−0.382 + 0.923i)15-s + (−0.866 − 0.499i)16-s + (0.793 + 0.608i)18-s − i·19-s + (0.707 − 0.707i)20-s + ⋯
L(s)  = 1  + (−0.608 − 0.793i)2-s + (−0.130 − 0.991i)3-s + (−0.258 + 0.965i)4-s + (−0.866 − 0.5i)5-s + (−0.707 + 0.707i)6-s + (0.923 − 0.382i)8-s + (−0.965 + 0.258i)9-s + (0.130 + 0.991i)10-s + (0.965 + 1.67i)11-s + (0.991 + 0.130i)12-s + (0.608 − 1.05i)13-s + (−0.382 + 0.923i)15-s + (−0.866 − 0.499i)16-s + (0.793 + 0.608i)18-s − i·19-s + (0.707 − 0.707i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3420\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $-0.766 + 0.642i$
Analytic conductor: \(1.70680\)
Root analytic conductor: \(1.30644\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3420} (1139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3420,\ (\ :0),\ -0.766 + 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7144826969\)
\(L(\frac12)\) \(\approx\) \(0.7144826969\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.608 + 0.793i)T \)
3 \( 1 + (0.130 + 0.991i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + iT \)
good7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.608 + 1.05i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - 1.58T + T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + 1.98iT - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-1.60 - 0.923i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.923 + 1.60i)T + (-0.5 + 0.866i)T^{2} \)
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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

−8.329437033946004577630034998495, −7.977783626921261244871734242584, −7.09964261463851479374254614600, −6.71400212403116049830554977192, −5.32518374129700463034654924745, −4.47544779888870334481435258210, −3.65973307474833938606233317041, −2.64118080410222518559570081112, −1.64017230237159332800273573674, −0.68178716466406545537556244799, 1.09137734718487361615315472795, 2.89146273401771508686794442121, 4.01623843812843241661853528292, 4.19673022273076374632566926147, 5.55729248504502775626724734506, 6.18180516558471899889199600136, 6.66375923387441076743584421219, 7.83900499708866855020984946155, 8.317175853571918987779862185328, 9.108759328793372962920640191392

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