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Properties

Label 1-3520-3520.1077-r0-0-0
Degree $1$
Conductor $3520$
Sign $0.991 - 0.133i$
Analytic cond. $16.3468$
Root an. cond. $16.3468$
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  + (−0.382 − 0.923i)3-s + (0.707 − 0.707i)7-s + (−0.707 + 0.707i)9-s + (0.923 − 0.382i)13-s − 17-s + (0.382 + 0.923i)19-s + (−0.923 − 0.382i)21-s + (0.707 + 0.707i)23-s + (0.923 + 0.382i)27-s + (−0.923 + 0.382i)29-s − 31-s + (0.923 + 0.382i)37-s + (−0.707 − 0.707i)39-s + (0.707 − 0.707i)41-s + (−0.382 + 0.923i)43-s + ⋯
L(s)  = 1  + (−0.382 − 0.923i)3-s + (0.707 − 0.707i)7-s + (−0.707 + 0.707i)9-s + (0.923 − 0.382i)13-s − 17-s + (0.382 + 0.923i)19-s + (−0.923 − 0.382i)21-s + (0.707 + 0.707i)23-s + (0.923 + 0.382i)27-s + (−0.923 + 0.382i)29-s − 31-s + (0.923 + 0.382i)37-s + (−0.707 − 0.707i)39-s + (0.707 − 0.707i)41-s + (−0.382 + 0.923i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.133i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.133i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $0.991 - 0.133i$
Analytic conductor: \(16.3468\)
Root analytic conductor: \(16.3468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3520} (1077, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3520,\ (0:\ ),\ 0.991 - 0.133i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.477312033 - 0.09886787230i\)
\(L(\frac12)\) \(\approx\) \(1.477312033 - 0.09886787230i\)
\(L(1)\) \(\approx\) \(0.9886562055 - 0.2382140982i\)
\(L(1)\) \(\approx\) \(0.9886562055 - 0.2382140982i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good3 \( 1 + (-0.382 - 0.923i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 + (0.923 - 0.382i)T \)
17 \( 1 - T \)
19 \( 1 + (0.382 + 0.923i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
29 \( 1 + (-0.923 + 0.382i)T \)
31 \( 1 - T \)
37 \( 1 + (0.923 + 0.382i)T \)
41 \( 1 + (0.707 - 0.707i)T \)
43 \( 1 + (-0.382 + 0.923i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.382 + 0.923i)T \)
59 \( 1 + (0.382 - 0.923i)T \)
61 \( 1 + (-0.923 + 0.382i)T \)
67 \( 1 + (0.382 + 0.923i)T \)
71 \( 1 + (0.707 + 0.707i)T \)
73 \( 1 + (0.707 + 0.707i)T \)
79 \( 1 + iT \)
83 \( 1 + (-0.923 + 0.382i)T \)
89 \( 1 + (0.707 + 0.707i)T \)
97 \( 1 - iT \)
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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

−18.43381789922412915915847080701, −18.14047166022530886194739276905, −17.33655711520933625440087489110, −16.63222593556667363352724167309, −15.97081960522547692209635905085, −15.25803331040365576700342557164, −14.9059266222465796282719372148, −14.03041343508900249914425356986, −13.230528924739741420654907954577, −12.40812957020633198441784510240, −11.43393257819118471949057605840, −11.17157421182650215895137984808, −10.60708043938654232356754903557, −9.3527774443346894814795543425, −9.08880993512597782337937941524, −8.42575288669983136628615238071, −7.39303050274142094168531282169, −6.42076319736247433557665855097, −5.80579145938757213865376643622, −4.98620830638659770776506843242, −4.43690085049331413333046977771, −3.62120422780007682914142098866, −2.667296651809832818014704481761, −1.83083500148928859347963972615, −0.545635013538687303018680126028, 0.94710609975818768052291465796, 1.48515662138657502360745489234, 2.387566115015819193366010057802, 3.48518959171605618671119577377, 4.2508873904177991937609678267, 5.29069594940912457551114817198, 5.80426296154630021733729332906, 6.715353165025619976311199270079, 7.39671878767153845021759967702, 7.95213394169690139085473464674, 8.67786804545915846961609269832, 9.553376135650951242051859285265, 10.7311594488579428432677720660, 11.06043133322758929926607634648, 11.61061018862560692803873581115, 12.68776594292738192663944176936, 13.09479054176164099359816344198, 13.83398952510178620230040970489, 14.37740129546800420270760642966, 15.25347980760521461900772847187, 16.11237296378476842888557968989, 16.88846248768406261067810083630, 17.348088194078281184142620147568, 18.22718550677427053553084230725, 18.40328642613431108798306996439

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