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Properties

Label 2-3420-285.227-c0-0-6
Degree $2$
Conductor $3420$
Sign $-0.318 + 0.948i$
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 + (−0.366 − 0.366i)7-s − 1.93i·11-s + (−1.22 + 1.22i)17-s − i·19-s + (−1.41 − 1.41i)23-s + (−0.866 − 0.499i)25-s + (0.448 − 0.258i)35-s + (0.366 − 0.366i)43-s + (−0.707 + 0.707i)47-s − 0.732i·49-s + (1.86 + 0.499i)55-s − 61-s + (1.36 − 1.36i)73-s + (−0.707 + 0.707i)77-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)5-s + (−0.366 − 0.366i)7-s − 1.93i·11-s + (−1.22 + 1.22i)17-s − i·19-s + (−1.41 − 1.41i)23-s + (−0.866 − 0.499i)25-s + (0.448 − 0.258i)35-s + (0.366 − 0.366i)43-s + (−0.707 + 0.707i)47-s − 0.732i·49-s + (1.86 + 0.499i)55-s − 61-s + (1.36 − 1.36i)73-s + (−0.707 + 0.707i)77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.318 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.318 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3420\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $-0.318 + 0.948i$
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: \((2,\ 3420,\ (\ :0),\ -0.318 + 0.948i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6049202873\)
\(L(\frac12)\) \(\approx\) \(0.6049202873\)
\(L(1)\) 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 \( 1 + (0.366 + 0.366i)T + iT^{2} \)
11 \( 1 + 1.93iT - T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
23 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
47 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
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

−8.431298193303242025865280019640, −7.975093523837504091580740690969, −6.86424925471777761059481014196, −6.34155196996909666895815618125, −5.88481643016533964258852876278, −4.51839790805903713915146442093, −3.77271380439567596863926266422, −3.04867160493463978979527672956, −2.13717974467701869231200817384, −0.33661159443476198702808413701, 1.60255027500122895457195215990, 2.36248264167170182165436540184, 3.73002746503298045970271074475, 4.45847986471689192288911696681, 5.09706881974316015407828603059, 5.92977610891591560523447598281, 6.88543791615224924870088865801, 7.58542541400586705024479083324, 8.182545373131979351682439216477, 9.241238874738344080503657231164

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