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Properties

Label 2-420-140.139-c1-0-17
Degree $2$
Conductor $420$
Sign $0.301 - 0.953i$
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  + (−1.33 + 0.475i)2-s + i·3-s + (1.54 − 1.26i)4-s + (2.22 − 0.260i)5-s + (−0.475 − 1.33i)6-s + (1.17 + 2.36i)7-s + (−1.45 + 2.42i)8-s − 9-s + (−2.83 + 1.40i)10-s − 0.365i·11-s + (1.26 + 1.54i)12-s + 1.72·13-s + (−2.69 − 2.59i)14-s + (0.260 + 2.22i)15-s + (0.790 − 3.92i)16-s − 0.146·17-s + ⋯
L(s)  = 1  + (−0.941 + 0.336i)2-s + 0.577i·3-s + (0.773 − 0.633i)4-s + (0.993 − 0.116i)5-s + (−0.194 − 0.543i)6-s + (0.445 + 0.895i)7-s + (−0.515 + 0.856i)8-s − 0.333·9-s + (−0.896 + 0.443i)10-s − 0.110i·11-s + (0.365 + 0.446i)12-s + 0.479·13-s + (−0.720 − 0.693i)14-s + (0.0672 + 0.573i)15-s + (0.197 − 0.980i)16-s − 0.0356·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.301 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.301 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.301 - 0.953i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ 0.301 - 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.913252 + 0.668687i\)
\(L(\frac12)\) \(\approx\) \(0.913252 + 0.668687i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.33 - 0.475i)T \)
3 \( 1 - iT \)
5 \( 1 + (-2.22 + 0.260i)T \)
7 \( 1 + (-1.17 - 2.36i)T \)
good11 \( 1 + 0.365iT - 11T^{2} \)
13 \( 1 - 1.72T + 13T^{2} \)
17 \( 1 + 0.146T + 17T^{2} \)
19 \( 1 - 4.73T + 19T^{2} \)
23 \( 1 + 3.44T + 23T^{2} \)
29 \( 1 + 5.29T + 29T^{2} \)
31 \( 1 - 6.07T + 31T^{2} \)
37 \( 1 - 7.52iT - 37T^{2} \)
41 \( 1 - 7.16iT - 41T^{2} \)
43 \( 1 + 8.65T + 43T^{2} \)
47 \( 1 - 3.83iT - 47T^{2} \)
53 \( 1 + 11.5iT - 53T^{2} \)
59 \( 1 - 6.18T + 59T^{2} \)
61 \( 1 + 0.348iT - 61T^{2} \)
67 \( 1 - 12.1T + 67T^{2} \)
71 \( 1 - 7.78iT - 71T^{2} \)
73 \( 1 + 4.55T + 73T^{2} \)
79 \( 1 + 14.2iT - 79T^{2} \)
83 \( 1 + 3.53iT - 83T^{2} \)
89 \( 1 + 13.6iT - 89T^{2} \)
97 \( 1 + 16.1T + 97T^{2} \)
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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

−11.28740610600701148572755474478, −10.04927078056040976396386634858, −9.676120801084029672176910922860, −8.697580275092725594138496431396, −8.065838068396673578688032670928, −6.58679804085972837703976183055, −5.74791991973640581522647754975, −4.98208528425008238155552150103, −2.94177368197966573992483964406, −1.62713276853245634864790635525, 1.12755614379170749935372102048, 2.25529905133244469154422408941, 3.71613306677079486171154328640, 5.49351309740286811030295227200, 6.61560900180680919000332545010, 7.39003125572785188852597450550, 8.269937390775121743804045091731, 9.304258715315951789920418082993, 10.11173117609980781289543027188, 10.87185024139084799045980860245

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