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Properties

Label 1-1375-1375.488-r1-0-0
Degree $1$
Conductor $1375$
Sign $-0.999 + 0.00135i$
Analytic cond. $147.764$
Root an. cond. $147.764$
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  + (0.248 − 0.968i)2-s + (−0.125 − 0.992i)3-s + (−0.876 − 0.481i)4-s + (−0.992 − 0.125i)6-s + (−0.951 − 0.309i)7-s + (−0.684 + 0.728i)8-s + (−0.968 + 0.248i)9-s + (−0.368 + 0.929i)12-s + (0.248 + 0.968i)13-s + (−0.535 + 0.844i)14-s + (0.535 + 0.844i)16-s + (0.684 − 0.728i)17-s + i·18-s + (0.187 + 0.982i)19-s + (−0.187 + 0.982i)21-s + ⋯
L(s)  = 1  + (0.248 − 0.968i)2-s + (−0.125 − 0.992i)3-s + (−0.876 − 0.481i)4-s + (−0.992 − 0.125i)6-s + (−0.951 − 0.309i)7-s + (−0.684 + 0.728i)8-s + (−0.968 + 0.248i)9-s + (−0.368 + 0.929i)12-s + (0.248 + 0.968i)13-s + (−0.535 + 0.844i)14-s + (0.535 + 0.844i)16-s + (0.684 − 0.728i)17-s + i·18-s + (0.187 + 0.982i)19-s + (−0.187 + 0.982i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.00135i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.00135i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $-0.999 + 0.00135i$
Analytic conductor: \(147.764\)
Root analytic conductor: \(147.764\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (488, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (1:\ ),\ -0.999 + 0.00135i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.0007708346783 - 1.134564751i\)
\(L(\frac12)\) \(\approx\) \(0.0007708346783 - 1.134564751i\)
\(L(1)\) \(\approx\) \(0.5919409807 - 0.6222553642i\)
\(L(1)\) \(\approx\) \(0.5919409807 - 0.6222553642i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.248 - 0.968i)T \)
3 \( 1 + (-0.125 - 0.992i)T \)
7 \( 1 + (-0.951 - 0.309i)T \)
13 \( 1 + (0.248 + 0.968i)T \)
17 \( 1 + (0.684 - 0.728i)T \)
19 \( 1 + (0.187 + 0.982i)T \)
23 \( 1 + (-0.248 + 0.968i)T \)
29 \( 1 + (0.992 - 0.125i)T \)
31 \( 1 + (-0.992 - 0.125i)T \)
37 \( 1 + (-0.998 - 0.0627i)T \)
41 \( 1 + (-0.929 - 0.368i)T \)
43 \( 1 + (0.951 + 0.309i)T \)
47 \( 1 + (0.481 - 0.876i)T \)
53 \( 1 + (-0.125 - 0.992i)T \)
59 \( 1 + (-0.968 + 0.248i)T \)
61 \( 1 + (0.535 - 0.844i)T \)
67 \( 1 + (-0.481 - 0.876i)T \)
71 \( 1 + (-0.187 + 0.982i)T \)
73 \( 1 + (0.844 + 0.535i)T \)
79 \( 1 + (-0.876 - 0.481i)T \)
83 \( 1 + (0.481 + 0.876i)T \)
89 \( 1 + (0.929 - 0.368i)T \)
97 \( 1 + (0.982 + 0.187i)T \)
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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

−21.285273610537492643019615206973, −20.39163083651696790377655046776, −19.54260736773753014444585993743, −18.61248866496217988052825100116, −17.71028439238581658622299281898, −17.05859116438903832924971644061, −16.26653299418647878413029811563, −15.73128602784184268651003800832, −15.16874781104529832221006196314, −14.40086985928344394959439597308, −13.547672128163032910461890915672, −12.6353952717955255511887001626, −12.07179680571725758500421891698, −10.65521172223260165222945479049, −10.1453096122877144792279472153, −9.14246465799644779229008582894, −8.66866001962716328687164454073, −7.692016749308316150046296657149, −6.58331666507459849906142561299, −5.90538701409763525205677798099, −5.22850150106663764508197449553, −4.31629087558231088961860568940, −3.39587103048516152140357733065, −2.82415811538515675399429046885, −0.648003461930461383118915177870, 0.33219807195577729521276436957, 1.333416260646692282966989450793, 2.13244093325813366214511216807, 3.2290096885067232977605690182, 3.8066707308257194324331789069, 5.14343706875257482114962671636, 5.91207863841256741039357377740, 6.77030541489001827443887782167, 7.64377179878668516511681368197, 8.694742064764688709582445822241, 9.493565941895726918393379026452, 10.238072932951874621174720975836, 11.205885445305026760782082726812, 12.01543315015903027783062026725, 12.39651247208361560775742108171, 13.35650275696992099429640042488, 13.90143410566433478872006360163, 14.41780966835006377387995455586, 15.79300273679031595303565107371, 16.66033912632911997976625721618, 17.4243316237807145536544174313, 18.41232209158791564264221140797, 18.82780768016523846230143453374, 19.47372535778086244660348952276, 20.14425351012671299923494605285

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