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Properties

Label 2-420-105.59-c1-0-0
Degree $2$
Conductor $420$
Sign $-0.979 + 0.200i$
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  + (−1.48 + 0.884i)3-s + (−1.06 + 1.96i)5-s + (0.0973 + 2.64i)7-s + (1.43 − 2.63i)9-s + (−2.62 − 1.51i)11-s + 2.31·13-s + (−0.149 − 3.87i)15-s + (−4.49 − 2.59i)17-s + (−5.58 + 3.22i)19-s + (−2.48 − 3.85i)21-s + (−2.43 − 4.21i)23-s + (−2.72 − 4.19i)25-s + (0.194 + 5.19i)27-s + 3.48i·29-s + (1.16 + 0.673i)31-s + ⋯
L(s)  = 1  + (−0.859 + 0.510i)3-s + (−0.477 + 0.878i)5-s + (0.0368 + 0.999i)7-s + (0.478 − 0.878i)9-s + (−0.792 − 0.457i)11-s + 0.642·13-s + (−0.0386 − 0.999i)15-s + (−1.09 − 0.629i)17-s + (−1.28 + 0.739i)19-s + (−0.542 − 0.840i)21-s + (−0.507 − 0.879i)23-s + (−0.544 − 0.838i)25-s + (0.0374 + 0.999i)27-s + 0.647i·29-s + (0.209 + 0.120i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.200i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.979 + 0.200i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.979 + 0.200i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ -0.979 + 0.200i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0340545 - 0.335690i\)
\(L(\frac12)\) \(\approx\) \(0.0340545 - 0.335690i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.48 - 0.884i)T \)
5 \( 1 + (1.06 - 1.96i)T \)
7 \( 1 + (-0.0973 - 2.64i)T \)
good11 \( 1 + (2.62 + 1.51i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.31T + 13T^{2} \)
17 \( 1 + (4.49 + 2.59i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.58 - 3.22i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.43 + 4.21i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.48iT - 29T^{2} \)
31 \( 1 + (-1.16 - 0.673i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.43 - 1.40i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 1.06T + 41T^{2} \)
43 \( 1 - 2.42iT - 43T^{2} \)
47 \( 1 + (3.30 - 1.90i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.52 + 6.11i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.18 - 10.7i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-11.4 + 6.60i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.72 + 2.14i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.08iT - 71T^{2} \)
73 \( 1 + (0.0763 - 0.132i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.04 + 5.27i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.2iT - 83T^{2} \)
89 \( 1 + (-7.10 - 12.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 8.63T + 97T^{2} \)
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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

−11.49356528118067703252261650117, −10.82239047042636653404410522986, −10.21369562670077981429041766415, −8.931467233158557315804360069795, −8.122525153269180284690034542835, −6.67384945244767698849766136048, −6.11148207144987512057069208718, −4.97714674955867394105138392004, −3.81695418265226224662508568167, −2.50146546090033568157148042284, 0.23126484144467665767489272828, 1.83481025487979681958153746803, 4.07220052472856640035741193786, 4.75806779395858771413182858317, 5.96908975126477328174116505956, 6.99957856210429350797425010154, 7.85313853147857516600549054093, 8.690808832682292006903091620238, 10.06670012834601994079291998474, 10.88907389609599160486880778770

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