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Properties

Label 1-1375-1375.767-r0-0-0
Degree $1$
Conductor $1375$
Sign $0.866 + 0.498i$
Analytic cond. $6.38547$
Root an. cond. $6.38547$
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  + (0.982 + 0.187i)2-s + (−0.998 − 0.0627i)3-s + (0.929 + 0.368i)4-s + (−0.968 − 0.248i)6-s + (0.587 + 0.809i)7-s + (0.844 + 0.535i)8-s + (0.992 + 0.125i)9-s + (−0.904 − 0.425i)12-s + (0.125 − 0.992i)13-s + (0.425 + 0.904i)14-s + (0.728 + 0.684i)16-s + (0.998 − 0.0627i)17-s + (0.951 + 0.309i)18-s + (0.968 + 0.248i)19-s + (−0.535 − 0.844i)21-s + ⋯
L(s)  = 1  + (0.982 + 0.187i)2-s + (−0.998 − 0.0627i)3-s + (0.929 + 0.368i)4-s + (−0.968 − 0.248i)6-s + (0.587 + 0.809i)7-s + (0.844 + 0.535i)8-s + (0.992 + 0.125i)9-s + (−0.904 − 0.425i)12-s + (0.125 − 0.992i)13-s + (0.425 + 0.904i)14-s + (0.728 + 0.684i)16-s + (0.998 − 0.0627i)17-s + (0.951 + 0.309i)18-s + (0.968 + 0.248i)19-s + (−0.535 − 0.844i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.866 + 0.498i$
Analytic conductor: \(6.38547\)
Root analytic conductor: \(6.38547\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (767, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ 0.866 + 0.498i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.537665547 + 0.6776868496i\)
\(L(\frac12)\) \(\approx\) \(2.537665547 + 0.6776868496i\)
\(L(1)\) \(\approx\) \(1.673671226 + 0.2917535445i\)
\(L(1)\) \(\approx\) \(1.673671226 + 0.2917535445i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.982 + 0.187i)T \)
3 \( 1 + (-0.998 - 0.0627i)T \)
7 \( 1 + (0.587 + 0.809i)T \)
13 \( 1 + (0.125 - 0.992i)T \)
17 \( 1 + (0.998 - 0.0627i)T \)
19 \( 1 + (0.968 + 0.248i)T \)
23 \( 1 + (-0.684 - 0.728i)T \)
29 \( 1 + (-0.637 + 0.770i)T \)
31 \( 1 + (0.535 - 0.844i)T \)
37 \( 1 + (0.125 - 0.992i)T \)
41 \( 1 + (0.992 + 0.125i)T \)
43 \( 1 + (0.587 - 0.809i)T \)
47 \( 1 + (0.248 + 0.968i)T \)
53 \( 1 + (-0.248 - 0.968i)T \)
59 \( 1 + (-0.876 + 0.481i)T \)
61 \( 1 + (-0.876 - 0.481i)T \)
67 \( 1 + (-0.998 + 0.0627i)T \)
71 \( 1 + (0.0627 - 0.998i)T \)
73 \( 1 + (-0.982 - 0.187i)T \)
79 \( 1 + (0.535 + 0.844i)T \)
83 \( 1 + (0.844 + 0.535i)T \)
89 \( 1 + (0.425 + 0.904i)T \)
97 \( 1 + (0.368 - 0.929i)T \)
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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

−21.035017176022359472883419808265, −20.286374262370676152521467517869, −19.34866039334896296548659248325, −18.54976482860526700065649436620, −17.5925692611455371705511117023, −16.82962156559048973224181325131, −16.23022490922136485379327799341, −15.51492321173071706691302234025, −14.49756065452979024259017541256, −13.82579761683810150945543595962, −13.206294156379330239557522545899, −11.99469938357902020925607630662, −11.78113104609870316456008267370, −10.94468767757887917835066282855, −10.21688419314431779939516417402, −9.46104227343223999801756390948, −7.736143823432713102900604153625, −7.28472303248803679199733396051, −6.28851445028951895841883392707, −5.60710177487900613534063375203, −4.69457937282625941128981990423, −4.156270392478970253226073506, −3.19865955670513344814265615406, −1.717037803948439151581094941204, −1.07811982591381663460687396663, 1.08944103251455289409251258634, 2.172804535936626425914650798871, 3.22371694696817121384323357662, 4.260944279093021849717124052898, 5.18323492751585169319305049532, 5.69193816974955761012089366894, 6.25826095112042211835922338812, 7.602697229439436754182558716738, 7.84396711510184647779446899292, 9.31715337875811564924485237957, 10.44996743138043681847934565966, 11.020213094050285107886090565061, 11.97383886301788192146759350564, 12.30530543246259898562561396299, 13.060754970292145870926382318742, 14.10917069735419591918093016652, 14.80092030647759453096082483132, 15.64151281580586026017837571732, 16.19129998794400738248620163270, 16.97240754051943930364826547707, 17.8819524665746889714094929180, 18.395254464739801984093117309587, 19.41937618160849040837268432526, 20.626815710573355289360335354965, 20.94908850994892346031031591203

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