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Properties

Label 1-1305-1305.1228-r0-0-0
Degree $1$
Conductor $1305$
Sign $-0.393 - 0.919i$
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  + (0.826 − 0.563i)2-s + (0.365 − 0.930i)4-s + (0.930 − 0.365i)7-s + (−0.222 − 0.974i)8-s + (0.680 − 0.733i)11-s + (−0.294 − 0.955i)13-s + (0.563 − 0.826i)14-s + (−0.733 − 0.680i)16-s + 17-s + (0.781 − 0.623i)19-s + (0.149 − 0.988i)22-s + (−0.563 + 0.826i)23-s + (−0.781 − 0.623i)26-s − i·28-s + (−0.997 + 0.0747i)31-s + (−0.988 − 0.149i)32-s + ⋯
L(s)  = 1  + (0.826 − 0.563i)2-s + (0.365 − 0.930i)4-s + (0.930 − 0.365i)7-s + (−0.222 − 0.974i)8-s + (0.680 − 0.733i)11-s + (−0.294 − 0.955i)13-s + (0.563 − 0.826i)14-s + (−0.733 − 0.680i)16-s + 17-s + (0.781 − 0.623i)19-s + (0.149 − 0.988i)22-s + (−0.563 + 0.826i)23-s + (−0.781 − 0.623i)26-s − i·28-s + (−0.997 + 0.0747i)31-s + (−0.988 − 0.149i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.393 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.393 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1305\)    =    \(3^{2} \cdot 5 \cdot 29\)
Sign: $-0.393 - 0.919i$
Analytic conductor: \(6.06039\)
Root analytic conductor: \(6.06039\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1305} (1228, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1305,\ (0:\ ),\ -0.393 - 0.919i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.626882299 - 2.465175521i\)
\(L(\frac12)\) \(\approx\) \(1.626882299 - 2.465175521i\)
\(L(1)\) \(\approx\) \(1.594717447 - 1.039647592i\)
\(L(1)\) \(\approx\) \(1.594717447 - 1.039647592i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.826 - 0.563i)T \)
7 \( 1 + (0.930 - 0.365i)T \)
11 \( 1 + (0.680 - 0.733i)T \)
13 \( 1 + (-0.294 - 0.955i)T \)
17 \( 1 + T \)
19 \( 1 + (0.781 - 0.623i)T \)
23 \( 1 + (-0.563 + 0.826i)T \)
31 \( 1 + (-0.997 + 0.0747i)T \)
37 \( 1 + (0.222 + 0.974i)T \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 + (-0.0747 + 0.997i)T \)
47 \( 1 + (0.733 + 0.680i)T \)
53 \( 1 + (0.433 - 0.900i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (-0.930 + 0.365i)T \)
67 \( 1 + (-0.680 - 0.733i)T \)
71 \( 1 + (0.222 - 0.974i)T \)
73 \( 1 + (-0.900 + 0.433i)T \)
79 \( 1 + (0.294 - 0.955i)T \)
83 \( 1 + (-0.149 - 0.988i)T \)
89 \( 1 + (-0.433 + 0.900i)T \)
97 \( 1 + (0.988 - 0.149i)T \)
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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

−21.36384749172257274626592356481, −20.576030583889378220842609596502, −20.044630081973857023623274194362, −18.72789285987634528021357655492, −18.12086480061847194034067910344, −17.10866242285226054111624019237, −16.67977879169414154177909443516, −15.76404322704737188649402972794, −14.8558541272191225928160518882, −14.32596721907496900348060298493, −13.90746481627661515561330149479, −12.55637705532696721573896573198, −12.057195708296234906481818371647, −11.53060456652206369360340622170, −10.39242105548830171566006406237, −9.276223018757250050939547900574, −8.49536616409327779584474197381, −7.52485796722634437254815496380, −7.0233962543434556194219815661, −5.88014184938802731483361211822, −5.23781241744029110547458134811, −4.31383897418680411869598437067, −3.674681835459700098268812848856, −2.33664270646661334509828114182, −1.61230448906814746484550789930, 0.92096165164712367909975987695, 1.63417110961919019821563052468, 2.975755668885752833036750732998, 3.56460402755879604382448516809, 4.62813088215828388395922492101, 5.37852423660550402293735614363, 6.06458709486564125207827096408, 7.26267385594088833787586811173, 7.95816584069853063299219390337, 9.14003821567672682782616584565, 10.03066040787216053124061111223, 10.76887834047391697786487165587, 11.62223644026786507604213085428, 11.996581073271683635692017859333, 13.17885287251925121698403345041, 13.736027995575173557863708790406, 14.54126417624831970515490800036, 15.034825128972206831941038471568, 16.04660693614041600147753875922, 16.86524116750774365947709258044, 17.843117716366752262714491746150, 18.509190151382892963811370828118, 19.565416310902659658741717800644, 20.02939785545710230229396059303, 20.77541456401793390306609638705

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