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Properties

Label 1-1305-1305.1003-r0-0-0
Degree $1$
Conductor $1305$
Sign $0.998 - 0.0456i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0456i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0456i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1305\)    =    \(3^{2} \cdot 5 \cdot 29\)
Sign: $0.998 - 0.0456i$
Analytic conductor: \(6.06039\)
Root analytic conductor: \(6.06039\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1305} (1003, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1305,\ (0:\ ),\ 0.998 - 0.0456i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.805911706 - 0.04126851355i\)
\(L(\frac12)\) \(\approx\) \(1.805911706 - 0.04126851355i\)
\(L(1)\) \(\approx\) \(1.277850647 - 0.3643970967i\)
\(L(1)\) \(\approx\) \(1.277850647 - 0.3643970967i\)

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

−21.222488567576807459370751253537, −20.21100551699960345064205509671, −19.62344361021297567017300070275, −18.44775481316350499942339534178, −17.53098069035325979194545464344, −17.257526493785185675143814907823, −16.44352477258352362520082342744, −15.47064777226484081449804311078, −14.77641845367472907075799364102, −14.28444300561215008221207423053, −13.29622554182721277896376783799, −12.85901770957688783254919359433, −11.418709065290588995867062567, −11.36286567056012298364635283814, −9.86584096120730692989200495102, −8.950975210232401488819801753063, −8.30012124540619662903229113075, −7.275179730652903642735872637021, −6.86553463007509274845125815538, −5.72395291907003113217775962180, −4.91221523548268543444056120906, −4.247793235504457186852842422579, −3.290640921221749749198174966195, −2.159972284633674645472665872307, −0.62622239904498430914383038306, 1.27858900898958147387101185252, 2.031446358885491632257376260935, 2.85978469899088731034390189933, 4.19978974084959401080918467303, 4.58206311160909313918584843602, 5.56542030641048196727789218148, 6.4828343685332549593897568040, 7.48753412218964211323878195617, 8.73844207777549379715842319653, 9.240182914829921572469929556745, 10.14920801461415106625858482268, 11.11491461654829981202855556140, 11.649201048567111168087991727506, 12.40955960187989061931755507901, 13.034859843406512620770211229161, 14.30794325234866929536071238478, 14.536635858886282043453805054914, 15.228210663585594169484030024270, 16.41937793245266447335856115542, 17.41707236339684476145589893371, 18.01816031513781762904164898007, 18.85157915268259152067459682810, 19.57428217725722574372860554059, 20.27548334891866424661915501270, 20.98736439768385362037667619480

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