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Properties

Label 1-407-407.327-r0-0-0
Degree $1$
Conductor $407$
Sign $0.583 - 0.811i$
Analytic cond. $1.89010$
Root an. cond. $1.89010$
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  + (−0.951 − 0.309i)2-s + (0.809 − 0.587i)3-s + (0.809 + 0.587i)4-s + (0.951 − 0.309i)5-s + (−0.951 + 0.309i)6-s + (0.809 + 0.587i)7-s + (−0.587 − 0.809i)8-s + (0.309 − 0.951i)9-s − 10-s + 12-s + (0.951 + 0.309i)13-s + (−0.587 − 0.809i)14-s + (0.587 − 0.809i)15-s + (0.309 + 0.951i)16-s + (−0.951 + 0.309i)17-s + (−0.587 + 0.809i)18-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)2-s + (0.809 − 0.587i)3-s + (0.809 + 0.587i)4-s + (0.951 − 0.309i)5-s + (−0.951 + 0.309i)6-s + (0.809 + 0.587i)7-s + (−0.587 − 0.809i)8-s + (0.309 − 0.951i)9-s − 10-s + 12-s + (0.951 + 0.309i)13-s + (−0.587 − 0.809i)14-s + (0.587 − 0.809i)15-s + (0.309 + 0.951i)16-s + (−0.951 + 0.309i)17-s + (−0.587 + 0.809i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.583 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.583 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $0.583 - 0.811i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (327, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ 0.583 - 0.811i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.328576578 - 0.6811089672i\)
\(L(\frac12)\) \(\approx\) \(1.328576578 - 0.6811089672i\)
\(L(1)\) \(\approx\) \(1.111368822 - 0.3692501190i\)
\(L(1)\) \(\approx\) \(1.111368822 - 0.3692501190i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.951 - 0.309i)T \)
3 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (0.951 - 0.309i)T \)
7 \( 1 + (0.809 + 0.587i)T \)
13 \( 1 + (0.951 + 0.309i)T \)
17 \( 1 + (-0.951 + 0.309i)T \)
19 \( 1 + (-0.587 - 0.809i)T \)
23 \( 1 - iT \)
29 \( 1 + (-0.587 + 0.809i)T \)
31 \( 1 + (0.951 + 0.309i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + iT \)
47 \( 1 + (-0.809 + 0.587i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (-0.587 + 0.809i)T \)
61 \( 1 + (0.951 - 0.309i)T \)
67 \( 1 - T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (0.951 + 0.309i)T \)
83 \( 1 + (-0.309 - 0.951i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.951 - 0.309i)T \)
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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

−24.89810297064988468857812875313, −23.89961970782102157530817608743, −22.72094619210084564812904872791, −21.45054869552023321376059396339, −20.78481240792343301229782436096, −20.27735994748176642167839496489, −19.13780271204260136691193346538, −18.308157854943244465895996255033, −17.436317564746315315334972390307, −16.755204900548376636542546659715, −15.57582361830595324347466387157, −14.96140377015965809143967363224, −13.931833010334762503369005199903, −13.40948276361807459948507018840, −11.453271652690659582696785743570, −10.61479690388796728668100940310, −10.02680186481025194333809774353, −9.02270152255808141679846063704, −8.2710941039716627854300808621, −7.34762464894809190358388896945, −6.1886796947560598830468605097, −5.10394053176917895019207158359, −3.7144645991743530615398429754, −2.316903713031884508479032025221, −1.51914649706811906310513213584, 1.29339480682876797546219274828, 2.0309626057216155120006201813, 2.90645158572137586353405293516, 4.499425129302046445430943517466, 6.172588830954605709143866946904, 6.83769803072435926792522994634, 8.33745525692478968102644638699, 8.63772882079210553021238013387, 9.4415664949382673676446066996, 10.64311376621837275139314654942, 11.57013234154826434579764151203, 12.72063214985880900652487624303, 13.34774736523414506066598824240, 14.513893036597344499050843145030, 15.37712521880979854592508454702, 16.51002471480269793577496318860, 17.71321710278360931212055542594, 18.00452555338557143924713820855, 18.86060518467511472861189777704, 19.8180599752825961740883446702, 20.72048171199321404187403677205, 21.17607461613128215126009978681, 22.0390915610713620289576913583, 23.83777548221859108636965427795, 24.52385687589252724327317212245

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