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Properties

Label 1-1740-1740.1679-r1-0-0
Degree $1$
Conductor $1740$
Sign $0.514 + 0.857i$
Analytic cond. $186.988$
Root an. cond. $186.988$
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  + (0.623 + 0.781i)7-s + (0.974 − 0.222i)11-s + (−0.222 − 0.974i)13-s + i·17-s + (−0.781 − 0.623i)19-s + (0.900 + 0.433i)23-s + (0.433 + 0.900i)31-s + (−0.974 − 0.222i)37-s + i·41-s + (0.433 − 0.900i)43-s + (0.974 − 0.222i)47-s + (−0.222 + 0.974i)49-s + (−0.900 + 0.433i)53-s + 59-s + (−0.781 + 0.623i)61-s + ⋯
L(s)  = 1  + (0.623 + 0.781i)7-s + (0.974 − 0.222i)11-s + (−0.222 − 0.974i)13-s + i·17-s + (−0.781 − 0.623i)19-s + (0.900 + 0.433i)23-s + (0.433 + 0.900i)31-s + (−0.974 − 0.222i)37-s + i·41-s + (0.433 − 0.900i)43-s + (0.974 − 0.222i)47-s + (−0.222 + 0.974i)49-s + (−0.900 + 0.433i)53-s + 59-s + (−0.781 + 0.623i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1740 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1740 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1740\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 29\)
Sign: $0.514 + 0.857i$
Analytic conductor: \(186.988\)
Root analytic conductor: \(186.988\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1740} (1679, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1740,\ (1:\ ),\ 0.514 + 0.857i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.134963396 + 1.208695548i\)
\(L(\frac12)\) \(\approx\) \(2.134963396 + 1.208695548i\)
\(L(1)\) \(\approx\) \(1.217352414 + 0.1624678071i\)
\(L(1)\) \(\approx\) \(1.217352414 + 0.1624678071i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
29 \( 1 \)
good7 \( 1 + (0.623 + 0.781i)T \)
11 \( 1 + (0.974 - 0.222i)T \)
13 \( 1 + (-0.222 - 0.974i)T \)
17 \( 1 + iT \)
19 \( 1 + (-0.781 - 0.623i)T \)
23 \( 1 + (0.900 + 0.433i)T \)
31 \( 1 + (0.433 + 0.900i)T \)
37 \( 1 + (-0.974 - 0.222i)T \)
41 \( 1 + iT \)
43 \( 1 + (0.433 - 0.900i)T \)
47 \( 1 + (0.974 - 0.222i)T \)
53 \( 1 + (-0.900 + 0.433i)T \)
59 \( 1 + T \)
61 \( 1 + (-0.781 + 0.623i)T \)
67 \( 1 + (0.222 - 0.974i)T \)
71 \( 1 + (0.222 + 0.974i)T \)
73 \( 1 + (0.433 - 0.900i)T \)
79 \( 1 + (0.974 + 0.222i)T \)
83 \( 1 + (-0.623 + 0.781i)T \)
89 \( 1 + (0.433 + 0.900i)T \)
97 \( 1 + (0.781 + 0.623i)T \)
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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

−20.02507071361899945303946478832, −19.14559214092778515734627266788, −18.67219785488973336694867135549, −17.5118777693936612326470782700, −17.079814592668064134003343829413, −16.47237405714274889481381015368, −15.5041555213976782018600752596, −14.45080411867090506119109217260, −14.26279758923850952603546636801, −13.36917701620022979435719113595, −12.38366680945909287404059819186, −11.64601976149415086986921679798, −11.03565842729613416181147515605, −10.14340668790070170205182528111, −9.31088630969180801767908528705, −8.62982588848774281562934466819, −7.59635999236704275326456051798, −6.94364179240390795717403048294, −6.244593044624028346058085298055, −4.98392047242318554354136445711, −4.36204854265912151172634397180, −3.63662029653142402197719133070, −2.350682127698635437887802073204, −1.492968218665016523512678060139, −0.52507922696668307500166889697, 0.89264114869670774951891695965, 1.80929726056990111428100229940, 2.79386108195862357373177488210, 3.70844753434555227779736035753, 4.718785163368031783896944640761, 5.483558467355182213187736741557, 6.28691801652361959942230795127, 7.13993362317173801959319339561, 8.20610119595658579197428977099, 8.716075771153795769008891857470, 9.46594979945338620341906317183, 10.61736482104662388245333265035, 11.07024887839879926351019791264, 12.12020813019716499711822765557, 12.54595896844881794600519076556, 13.52442049907425800523050894447, 14.38224579705892025620508122531, 15.17014474555029531196944661565, 15.43766677604021841674053504987, 16.67099068768853063670949157214, 17.44466366880883583565161405141, 17.745024199377033066579140692504, 18.918498587510935169049461624971, 19.37638868929812309507094118791, 20.14052434526934470064741282450

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