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Properties

Label 1-3525-3525.11-r0-0-0
Degree $1$
Conductor $3525$
Sign $0.566 + 0.824i$
Analytic cond. $16.3700$
Root an. cond. $16.3700$
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  + (0.969 + 0.243i)2-s + (0.881 + 0.472i)4-s + (0.682 − 0.730i)7-s + (0.740 + 0.672i)8-s + (−0.662 + 0.749i)11-s + (0.702 − 0.711i)13-s + (0.839 − 0.542i)14-s + (0.554 + 0.832i)16-s + (−0.507 + 0.861i)17-s + (0.0136 + 0.999i)19-s + (−0.824 + 0.565i)22-s + (0.927 + 0.373i)23-s + (0.854 − 0.519i)26-s + (0.946 − 0.321i)28-s + (−0.894 + 0.447i)29-s + ⋯
L(s)  = 1  + (0.969 + 0.243i)2-s + (0.881 + 0.472i)4-s + (0.682 − 0.730i)7-s + (0.740 + 0.672i)8-s + (−0.662 + 0.749i)11-s + (0.702 − 0.711i)13-s + (0.839 − 0.542i)14-s + (0.554 + 0.832i)16-s + (−0.507 + 0.861i)17-s + (0.0136 + 0.999i)19-s + (−0.824 + 0.565i)22-s + (0.927 + 0.373i)23-s + (0.854 − 0.519i)26-s + (0.946 − 0.321i)28-s + (−0.894 + 0.447i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.566 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.566 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $0.566 + 0.824i$
Analytic conductor: \(16.3700\)
Root analytic conductor: \(16.3700\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3525,\ (0:\ ),\ 0.566 + 0.824i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.368369711 + 1.773235762i\)
\(L(\frac12)\) \(\approx\) \(3.368369711 + 1.773235762i\)
\(L(1)\) \(\approx\) \(2.060399590 + 0.4971861140i\)
\(L(1)\) \(\approx\) \(2.060399590 + 0.4971861140i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
47 \( 1 \)
good2 \( 1 + (0.969 + 0.243i)T \)
7 \( 1 + (0.682 - 0.730i)T \)
11 \( 1 + (-0.662 + 0.749i)T \)
13 \( 1 + (0.702 - 0.711i)T \)
17 \( 1 + (-0.507 + 0.861i)T \)
19 \( 1 + (0.0136 + 0.999i)T \)
23 \( 1 + (0.927 + 0.373i)T \)
29 \( 1 + (-0.894 + 0.447i)T \)
31 \( 1 + (0.620 - 0.784i)T \)
37 \( 1 + (-0.839 - 0.542i)T \)
41 \( 1 + (0.994 - 0.109i)T \)
43 \( 1 + (0.990 - 0.136i)T \)
53 \( 1 + (0.740 - 0.672i)T \)
59 \( 1 + (0.176 + 0.984i)T \)
61 \( 1 + (-0.839 + 0.542i)T \)
67 \( 1 + (0.981 + 0.190i)T \)
71 \( 1 + (-0.641 + 0.767i)T \)
73 \( 1 + (0.230 + 0.973i)T \)
79 \( 1 + (-0.282 + 0.959i)T \)
83 \( 1 + (-0.976 + 0.216i)T \)
89 \( 1 + (0.955 - 0.295i)T \)
97 \( 1 + (-0.620 - 0.784i)T \)
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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

−18.84730651830370143659739017170, −18.07465559501076817001441071603, −17.22045011476056025845938245498, −16.228647717710562371034715543037, −15.74792686551787770497731935591, −15.19892481780911679819684528312, −14.36998846035255408611765168519, −13.69232517938317257305626887192, −13.27388483205866106056447738071, −12.36326899845984821905485193168, −11.66587375997275320954049480201, −11.01561165389719020296329492330, −10.75400554743220413313968936631, −9.38966668138458844267711848799, −8.86606134141595146585209179577, −7.945417165239001145239381969382, −7.06805643558664083492447653707, −6.349101516731120759014519593135, −5.57132553771606954651341956373, −4.90377206021137235434630745118, −4.345990132314906226920069697925, −3.19834543119620869716978843084, −2.639353361408240685102654714817, −1.844311671308125666899567527312, −0.804557666216515007755817450240, 1.18035406930569135013461943305, 1.974763581365754129358435251910, 2.86881830119659692409282422353, 3.97502398948345180446692936254, 4.16781698282809471672898817377, 5.36778767756058876472075415391, 5.66230640180532394577220656018, 6.76002903403037589533546635115, 7.451084206798498754249391493071, 7.9745486460313028379575084898, 8.73333811481141553394375419403, 10.02172942552979406968835047430, 10.72241138823818996750779743528, 11.09184861479903552177055381947, 12.04158454407266992954776513025, 12.9375157141455932071870279650, 13.12814159706089977836944918005, 14.055945636266471467840657511740, 14.70964311302793489709843650319, 15.2853277357367126793682707021, 15.87876972228710133245046275843, 16.7659187201189687605543391239, 17.3721647295085709849663655117, 17.91383323183816070344288297879, 18.88085968999983689661798209543

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