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Properties

Label 2-420-12.11-c1-0-18
Degree $2$
Conductor $420$
Sign $0.927 - 0.372i$
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  + (−0.0797 + 1.41i)2-s + (−1.52 + 0.822i)3-s + (−1.98 − 0.225i)4-s − i·5-s + (−1.03 − 2.21i)6-s − i·7-s + (0.476 − 2.78i)8-s + (1.64 − 2.50i)9-s + (1.41 + 0.0797i)10-s − 2.48·11-s + (3.21 − 1.29i)12-s + 5.62·13-s + (1.41 + 0.0797i)14-s + (0.822 + 1.52i)15-s + (3.89 + 0.895i)16-s + 0.700i·17-s + ⋯
L(s)  = 1  + (−0.0563 + 0.998i)2-s + (−0.880 + 0.474i)3-s + (−0.993 − 0.112i)4-s − 0.447i·5-s + (−0.424 − 0.905i)6-s − 0.377i·7-s + (0.168 − 0.985i)8-s + (0.549 − 0.835i)9-s + (0.446 + 0.0252i)10-s − 0.750·11-s + (0.927 − 0.372i)12-s + 1.56·13-s + (0.377 + 0.0213i)14-s + (0.212 + 0.393i)15-s + (0.974 + 0.223i)16-s + 0.169i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.372i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 - 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.927 - 0.372i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ 0.927 - 0.372i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.835123 + 0.161391i\)
\(L(\frac12)\) \(\approx\) \(0.835123 + 0.161391i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0797 - 1.41i)T \)
3 \( 1 + (1.52 - 0.822i)T \)
5 \( 1 + iT \)
7 \( 1 + iT \)
good11 \( 1 + 2.48T + 11T^{2} \)
13 \( 1 - 5.62T + 13T^{2} \)
17 \( 1 - 0.700iT - 17T^{2} \)
19 \( 1 + 4.84iT - 19T^{2} \)
23 \( 1 - 3.16T + 23T^{2} \)
29 \( 1 + 5.02iT - 29T^{2} \)
31 \( 1 - 4.00iT - 31T^{2} \)
37 \( 1 - 3.83T + 37T^{2} \)
41 \( 1 - 0.589iT - 41T^{2} \)
43 \( 1 + 10.9iT - 43T^{2} \)
47 \( 1 - 13.3T + 47T^{2} \)
53 \( 1 + 8.21iT - 53T^{2} \)
59 \( 1 - 8.14T + 59T^{2} \)
61 \( 1 + 15.3T + 61T^{2} \)
67 \( 1 + 2.48iT - 67T^{2} \)
71 \( 1 - 1.15T + 71T^{2} \)
73 \( 1 - 6.50T + 73T^{2} \)
79 \( 1 + 16.5iT - 79T^{2} \)
83 \( 1 + 3.32T + 83T^{2} \)
89 \( 1 - 1.26iT - 89T^{2} \)
97 \( 1 - 8.03T + 97T^{2} \)
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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

−11.00255324051265467199850964661, −10.39673085265860159894611611010, −9.266803990143340307607636937218, −8.558897762542468658614715817173, −7.39176505602234449826239549571, −6.42173694370493109351357293515, −5.56168571301585039860530015153, −4.69357987735343933058725055305, −3.70813651393126171160580130160, −0.75174178162191923021440709877, 1.30302209213817162326411786732, 2.75471068388549638944159541282, 4.11238174426060265809156697992, 5.42583453331924379370951671717, 6.15545391475698986823261952300, 7.56787838522252346073741578933, 8.440260700235445813854005646715, 9.601372832899930279761345171585, 10.74541559471796127163431611268, 10.94836699028963075841267730761

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