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Properties

Label 1-1305-1305.1177-r0-0-0
Degree $1$
Conductor $1305$
Sign $0.944 - 0.328i$
Analytic cond. $6.06039$
Root an. cond. $6.06039$
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  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s + (0.866 − 0.5i)7-s + 8-s + (−0.866 + 0.5i)11-s + (−0.866 − 0.5i)13-s + (−0.866 − 0.5i)14-s + (−0.5 − 0.866i)16-s + 17-s + i·19-s + (0.866 + 0.5i)22-s + (0.866 + 0.5i)23-s + i·26-s + i·28-s + (0.866 + 0.5i)31-s + (−0.5 + 0.866i)32-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s + (0.866 − 0.5i)7-s + 8-s + (−0.866 + 0.5i)11-s + (−0.866 − 0.5i)13-s + (−0.866 − 0.5i)14-s + (−0.5 − 0.866i)16-s + 17-s + i·19-s + (0.866 + 0.5i)22-s + (0.866 + 0.5i)23-s + i·26-s + i·28-s + (0.866 + 0.5i)31-s + (−0.5 + 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1305\)    =    \(3^{2} \cdot 5 \cdot 29\)
Sign: $0.944 - 0.328i$
Analytic conductor: \(6.06039\)
Root analytic conductor: \(6.06039\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1305} (1177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1305,\ (0:\ ),\ 0.944 - 0.328i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.090558725 - 0.1842900445i\)
\(L(\frac12)\) \(\approx\) \(1.090558725 - 0.1842900445i\)
\(L(1)\) \(\approx\) \(0.8179429913 - 0.2402011240i\)
\(L(1)\) \(\approx\) \(0.8179429913 - 0.2402011240i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (-0.866 - 0.5i)T \)
17 \( 1 + T \)
19 \( 1 + iT \)
23 \( 1 + (0.866 + 0.5i)T \)
31 \( 1 + (0.866 + 0.5i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.866 - 0.5i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 - iT \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (-0.866 + 0.5i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 - T \)
73 \( 1 + T \)
79 \( 1 + (0.866 - 0.5i)T \)
83 \( 1 + (-0.866 + 0.5i)T \)
89 \( 1 + iT \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−21.10827346861415081269195967849, −20.1645529721755204894363196840, −19.06043581959967717338477929386, −18.75305956222330813554006369941, −17.92311887925895499322570648579, −17.087733878306116079964156251301, −16.642637748395663192802640668383, −15.421054282954056441760911808741, −15.220855720966150661594496758828, −14.17845463302879824967669949833, −13.66978157136089797071482273406, −12.54321472108791343748677241687, −11.60396202868781763902900051198, −10.73326551661366746028882446130, −9.99146034945091048385894214150, −8.99846379047098246849372656524, −8.418636896880097590567202142167, −7.57872163728390980022283380583, −6.91956360447547889283228868627, −5.77181296534737649586224666351, −5.11603178202359522879753253534, −4.48969356981647767655047103151, −2.91559389771529198244607560046, −1.91755164408001581567048163946, −0.65843920232178808395064977393, 0.952614828099121477430460903544, 1.864482356853063622688241773071, 2.851614237917084067872512702613, 3.73075765239015800729404220539, 4.8451568816091454388965759716, 5.32998032150072703778418043640, 7.01994527475504137169027280259, 7.80291775802706685731957553674, 8.17584257327532876979633753290, 9.38606490949327878620938536187, 10.254903498000756201336715271375, 10.554852367625773349151991467439, 11.63964313979485827523090227784, 12.30464569214499238639129683187, 12.991548556208283813908883380263, 13.96144677139845957114556757405, 14.63158220385492458373522212167, 15.64440055768411087819309749761, 16.68267951908562788214158195033, 17.36720504269032651952812760436, 17.84756754200788445935759141741, 18.76634773667098063922516986183, 19.366087021433980216142788335107, 20.29621574497160350714286412340, 20.976189596435639922993236925552

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