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Properties

Label 2-420-60.59-c1-0-10
Degree $2$
Conductor $420$
Sign $-0.289 - 0.957i$
Analytic cond. $3.35371$
Root an. cond. $1.83131$
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  + 1.41i·2-s + (−1 − 1.41i)3-s − 2.00·4-s + (−2.12 − 0.707i)5-s + (2.00 − 1.41i)6-s + 7-s − 2.82i·8-s + (−1.00 + 2.82i)9-s + (1.00 − 3i)10-s + 4.24·11-s + (2.00 + 2.82i)12-s + 6i·13-s + 1.41i·14-s + (1.12 + 3.70i)15-s + 4.00·16-s − 4.24·17-s + ⋯
L(s)  = 1  + 0.999i·2-s + (−0.577 − 0.816i)3-s − 1.00·4-s + (−0.948 − 0.316i)5-s + (0.816 − 0.577i)6-s + 0.377·7-s − 1.00i·8-s + (−0.333 + 0.942i)9-s + (0.316 − 0.948i)10-s + 1.27·11-s + (0.577 + 0.816i)12-s + 1.66i·13-s + 0.377i·14-s + (0.289 + 0.957i)15-s + 1.00·16-s − 1.02·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.289 - 0.957i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ -0.289 - 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.436634 + 0.588244i\)
\(L(\frac12)\) \(\approx\) \(0.436634 + 0.588244i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
3 \( 1 + (1 + 1.41i)T \)
5 \( 1 + (2.12 + 0.707i)T \)
7 \( 1 - T \)
good11 \( 1 - 4.24T + 11T^{2} \)
13 \( 1 - 6iT - 13T^{2} \)
17 \( 1 + 4.24T + 17T^{2} \)
19 \( 1 - 6iT - 19T^{2} \)
23 \( 1 + 1.41iT - 23T^{2} \)
29 \( 1 - 2.82iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 + 1.41iT - 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 - 2.82iT - 47T^{2} \)
53 \( 1 - 8.48T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 12.7T + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 2.82iT - 83T^{2} \)
89 \( 1 - 7.07iT - 89T^{2} \)
97 \( 1 + 6iT - 97T^{2} \)
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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

−11.85071299796015118271473679414, −10.74012391998138545668401065977, −9.175707495155205436208128308999, −8.572364036456113606843158707769, −7.56804664903695418445123876235, −6.81150926493358275420333871334, −6.09455662564269842306988859088, −4.69318868881036345613453703348, −4.01801459508945364658875763736, −1.42475723401941157823212917631, 0.57625560195251881616498648229, 2.89216360770343845110556339535, 3.97144409121328041408280326096, 4.67353054722321219660327720565, 5.87594272878261745960226205192, 7.29140924592713371793219504879, 8.582571953545101254046710222439, 9.223931677752297718081719334088, 10.37880128520914666275638233163, 11.02864775612169548170545179400

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