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Properties

Label 2-420-28.27-c1-0-20
Degree $2$
Conductor $420$
Sign $-0.358 + 0.933i$
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  + (−0.449 − 1.34i)2-s + 3-s + (−1.59 + 1.20i)4-s + i·5-s + (−0.449 − 1.34i)6-s + (−1.40 − 2.24i)7-s + (2.33 + 1.59i)8-s + 9-s + (1.34 − 0.449i)10-s − 3.99i·11-s + (−1.59 + 1.20i)12-s − 2.50i·13-s + (−2.38 + 2.88i)14-s + i·15-s + (1.09 − 3.84i)16-s − 1.07i·17-s + ⋯
L(s)  = 1  + (−0.317 − 0.948i)2-s + 0.577·3-s + (−0.797 + 0.602i)4-s + 0.447i·5-s + (−0.183 − 0.547i)6-s + (−0.529 − 0.848i)7-s + (0.825 + 0.564i)8-s + 0.333·9-s + (0.424 − 0.142i)10-s − 1.20i·11-s + (−0.460 + 0.348i)12-s − 0.694i·13-s + (−0.636 + 0.771i)14-s + 0.258i·15-s + (0.273 − 0.961i)16-s − 0.261i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.358 + 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.358 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.358 + 0.933i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ -0.358 + 0.933i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.685850 - 0.997524i\)
\(L(\frac12)\) \(\approx\) \(0.685850 - 0.997524i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.449 + 1.34i)T \)
3 \( 1 - T \)
5 \( 1 - iT \)
7 \( 1 + (1.40 + 2.24i)T \)
good11 \( 1 + 3.99iT - 11T^{2} \)
13 \( 1 + 2.50iT - 13T^{2} \)
17 \( 1 + 1.07iT - 17T^{2} \)
19 \( 1 - 7.78T + 19T^{2} \)
23 \( 1 + 8.10iT - 23T^{2} \)
29 \( 1 + 5.99T + 29T^{2} \)
31 \( 1 - 6.32T + 31T^{2} \)
37 \( 1 - 3.05T + 37T^{2} \)
41 \( 1 - 9.32iT - 41T^{2} \)
43 \( 1 - 8.12iT - 43T^{2} \)
47 \( 1 + 9.01T + 47T^{2} \)
53 \( 1 - 2.86T + 53T^{2} \)
59 \( 1 + 9.68T + 59T^{2} \)
61 \( 1 - 3.21iT - 61T^{2} \)
67 \( 1 + 7.79iT - 67T^{2} \)
71 \( 1 - 5.84iT - 71T^{2} \)
73 \( 1 + 5.73iT - 73T^{2} \)
79 \( 1 + 2.81iT - 79T^{2} \)
83 \( 1 - 12.5T + 83T^{2} \)
89 \( 1 - 3.51iT - 89T^{2} \)
97 \( 1 - 12.0iT - 97T^{2} \)
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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

−10.85614315402036747474648590185, −10.02066817463839279405917421981, −9.379196114309090284132032988664, −8.197657278358360438805456242398, −7.58671458202983686105733960652, −6.27313934737801250667711288784, −4.70756630836556992238289291746, −3.37702357293819270061430931327, −2.90330909851266986174204021352, −0.883572526905719778457234256240, 1.78042606484901084746052171429, 3.62862494135058425722969852821, 4.92571976674751249450188500110, 5.79179664024840901015471195337, 7.05734319041761942910666226568, 7.69449410913779780875104499605, 8.839514954833766229809367352075, 9.503489568042012786740263323694, 9.920038055829274480374715013918, 11.61684079346044718709085990271

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