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Properties

Label 1-1205-1205.102-r0-0-0
Degree $1$
Conductor $1205$
Sign $-0.539 + 0.842i$
Analytic cond. $5.59599$
Root an. cond. $5.59599$
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  + (−0.707 − 0.707i)2-s + (0.987 + 0.156i)3-s + i·4-s + (−0.587 − 0.809i)6-s + (−0.0784 + 0.996i)7-s + (0.707 − 0.707i)8-s + (0.951 + 0.309i)9-s + (−0.382 + 0.923i)11-s + (−0.156 + 0.987i)12-s + (−0.522 + 0.852i)13-s + (0.760 − 0.649i)14-s − 16-s + (−0.649 − 0.760i)17-s + (−0.453 − 0.891i)18-s + (−0.923 − 0.382i)19-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + (0.987 + 0.156i)3-s + i·4-s + (−0.587 − 0.809i)6-s + (−0.0784 + 0.996i)7-s + (0.707 − 0.707i)8-s + (0.951 + 0.309i)9-s + (−0.382 + 0.923i)11-s + (−0.156 + 0.987i)12-s + (−0.522 + 0.852i)13-s + (0.760 − 0.649i)14-s − 16-s + (−0.649 − 0.760i)17-s + (−0.453 − 0.891i)18-s + (−0.923 − 0.382i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.539 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.539 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $-0.539 + 0.842i$
Analytic conductor: \(5.59599\)
Root analytic conductor: \(5.59599\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1205} (102, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1205,\ (0:\ ),\ -0.539 + 0.842i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3595189710 + 0.6572598531i\)
\(L(\frac12)\) \(\approx\) \(0.3595189710 + 0.6572598531i\)
\(L(1)\) \(\approx\) \(0.8272397140 + 0.09355102274i\)
\(L(1)\) \(\approx\) \(0.8272397140 + 0.09355102274i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (0.987 + 0.156i)T \)
7 \( 1 + (-0.0784 + 0.996i)T \)
11 \( 1 + (-0.382 + 0.923i)T \)
13 \( 1 + (-0.522 + 0.852i)T \)
17 \( 1 + (-0.649 - 0.760i)T \)
19 \( 1 + (-0.923 - 0.382i)T \)
23 \( 1 + (-0.972 + 0.233i)T \)
29 \( 1 + (-0.891 + 0.453i)T \)
31 \( 1 + (0.760 - 0.649i)T \)
37 \( 1 + (0.522 + 0.852i)T \)
41 \( 1 + (0.891 + 0.453i)T \)
43 \( 1 + (-0.996 + 0.0784i)T \)
47 \( 1 + (-0.453 - 0.891i)T \)
53 \( 1 + (-0.891 + 0.453i)T \)
59 \( 1 + (0.987 - 0.156i)T \)
61 \( 1 + (-0.987 - 0.156i)T \)
67 \( 1 + (-0.453 + 0.891i)T \)
71 \( 1 + (0.996 - 0.0784i)T \)
73 \( 1 + (-0.522 - 0.852i)T \)
79 \( 1 + (-0.987 + 0.156i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (0.382 + 0.923i)T \)
97 \( 1 + (0.309 - 0.951i)T \)
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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

−20.65302599321802813315793206384, −19.80936096785930484562536419887, −19.48374658911034073780755716156, −18.669274697754713379909977445959, −17.79200023818526769143707636096, −17.13040967626781729929446015244, −16.22958051942867863876463010969, −15.57168721966722027239677018815, −14.70681170115801386023310018290, −14.15314297028669842656381772779, −13.322915856796708267686905984692, −12.69302898258737181317364665513, −11.0998563354047015031873588617, −10.39831986523878133169035397925, −9.82951996230551544105125006614, −8.77534897031009530655478209851, −8.06870964112690365124296475834, −7.64821546661753840303979947732, −6.61757519872152648156036223655, −5.89150143492180982345813862043, −4.579947586821795468653056111386, −3.74831810997882167384911888243, −2.54874770223883790049471374790, −1.54428473727887900560233429006, −0.306369561139499165552191288916, 1.789354961683305511061591926606, 2.26764033220019527270272909095, 2.99682085096107834300228090620, 4.236525037483980681541820258174, 4.80899315792996856735110654261, 6.49796525526321284622582823477, 7.349948895377675338612495494903, 8.160986789957446728755588808372, 8.93151187006002208448020995928, 9.57343364323663488727533840075, 10.07082081775802915640587744821, 11.28812898662788619021700570309, 11.99121288076266152836154889504, 12.85388429842321864618018059732, 13.43421523279159351793582262029, 14.56556403463707483854381328693, 15.31584271686554563900170748237, 15.98562218871222980023522322560, 16.90654303267370215706874056220, 17.96359914737264978491773717750, 18.48909055024650032115461859451, 19.19124176612922852836965832877, 19.90965516499105957385601087280, 20.50454353834169729623730600301, 21.32456278635758606667993933301

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