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Properties

Label 1-1407-1407.1199-r1-0-0
Degree $1$
Conductor $1407$
Sign $0.992 + 0.125i$
Analytic cond. $151.203$
Root an. cond. $151.203$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.995 + 0.0950i)2-s + (0.981 + 0.189i)4-s + (0.786 − 0.618i)5-s + (0.959 + 0.281i)8-s + (0.841 − 0.540i)10-s + (0.786 − 0.618i)11-s + (0.723 + 0.690i)13-s + (0.928 + 0.371i)16-s + (0.654 + 0.755i)17-s + (0.580 + 0.814i)19-s + (0.888 − 0.458i)20-s + (0.841 − 0.540i)22-s + (0.888 − 0.458i)23-s + (0.235 − 0.971i)25-s + (0.654 + 0.755i)26-s + ⋯
L(s)  = 1  + (0.995 + 0.0950i)2-s + (0.981 + 0.189i)4-s + (0.786 − 0.618i)5-s + (0.959 + 0.281i)8-s + (0.841 − 0.540i)10-s + (0.786 − 0.618i)11-s + (0.723 + 0.690i)13-s + (0.928 + 0.371i)16-s + (0.654 + 0.755i)17-s + (0.580 + 0.814i)19-s + (0.888 − 0.458i)20-s + (0.841 − 0.540i)22-s + (0.888 − 0.458i)23-s + (0.235 − 0.971i)25-s + (0.654 + 0.755i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $0.992 + 0.125i$
Analytic conductor: \(151.203\)
Root analytic conductor: \(151.203\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (1199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1407,\ (1:\ ),\ 0.992 + 0.125i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(7.020861939 + 0.4406635869i\)
\(L(\frac12)\) \(\approx\) \(7.020861939 + 0.4406635869i\)
\(L(1)\) \(\approx\) \(2.757133557 + 0.05830051950i\)
\(L(1)\) \(\approx\) \(2.757133557 + 0.05830051950i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
67 \( 1 \)
good2 \( 1 + (0.995 + 0.0950i)T \)
5 \( 1 + (0.786 - 0.618i)T \)
11 \( 1 + (0.786 - 0.618i)T \)
13 \( 1 + (0.723 + 0.690i)T \)
17 \( 1 + (0.654 + 0.755i)T \)
19 \( 1 + (0.580 + 0.814i)T \)
23 \( 1 + (0.888 - 0.458i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (0.235 + 0.971i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.981 + 0.189i)T \)
43 \( 1 + (-0.654 - 0.755i)T \)
47 \( 1 + (-0.841 - 0.540i)T \)
53 \( 1 + (0.327 + 0.945i)T \)
59 \( 1 + (-0.723 + 0.690i)T \)
61 \( 1 + (-0.786 - 0.618i)T \)
71 \( 1 + (0.327 + 0.945i)T \)
73 \( 1 + (-0.142 + 0.989i)T \)
79 \( 1 + (-0.959 + 0.281i)T \)
83 \( 1 + (0.786 - 0.618i)T \)
89 \( 1 + (0.888 + 0.458i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.75023563714113511422604133351, −20.0398603434074216687374718019, −19.23805884843278618362597427020, −18.2641289157786788415446667327, −17.584768214573271100094445671320, −16.72083373888982849758598259365, −15.83487782632548965846058937756, −15.021598149549482431144526821310, −14.486740966142846638859200117916, −13.624578677094520585780943662002, −13.18675276169816124739391642939, −12.18839696550805382365960692464, −11.4064389073665362849745313240, −10.721061100394553627970481440503, −9.8585655050170216922185007787, −9.14081858331643378189577318260, −7.71724182576594297714918459504, −6.944528573270109557233478186453, −6.33025981779042677904891989095, −5.38637509700092536469928394564, −4.7821054045038228646996888283, −3.43498706758576846418576630536, −3.019089094922154347684886921887, −1.87412407859612800378966524022, −1.02229999295895293374896732527, 1.2033019124814703412261802065, 1.64685621049608221443464126666, 2.99964463040233437458144070516, 3.77710643284519498579937908319, 4.69397079054457575829533134219, 5.546811313306654270050518358135, 6.25161123123569696972496024371, 6.842320912093665887555556602209, 8.20474489044161685270869385297, 8.75649051308820921026277075775, 9.918981961205320290706419639220, 10.63080137548675541677504201283, 11.72662709104656553040439366991, 12.174832175037656259846199931316, 13.12445661579341765332103080830, 13.832133318165359717340102711854, 14.224738597896023316275261256551, 15.18845777392776292800745589647, 16.11500572424538653975690842734, 16.841337143474749907573088940796, 17.11275513986098414206147485955, 18.46963791016582172317801686179, 19.20509797095585640308001167292, 20.15510192226121587466720297819, 20.805150015708948457862285644960

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