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Properties

Label 1-2952-2952.1739-r1-0-0
Degree $1$
Conductor $2952$
Sign $-0.931 - 0.363i$
Analytic cond. $317.236$
Root an. cond. $317.236$
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  + (−0.994 + 0.104i)5-s + (−0.544 + 0.838i)7-s + (−0.629 − 0.777i)11-s + (−0.838 + 0.544i)13-s + (−0.156 − 0.987i)17-s + (−0.891 + 0.453i)19-s + (0.978 + 0.207i)23-s + (0.978 − 0.207i)25-s + (−0.933 + 0.358i)29-s + (0.913 − 0.406i)31-s + (0.453 − 0.891i)35-s + (0.809 − 0.587i)37-s + (0.743 + 0.669i)43-s + (−0.998 + 0.0523i)47-s + (−0.406 − 0.913i)49-s + ⋯
L(s)  = 1  + (−0.994 + 0.104i)5-s + (−0.544 + 0.838i)7-s + (−0.629 − 0.777i)11-s + (−0.838 + 0.544i)13-s + (−0.156 − 0.987i)17-s + (−0.891 + 0.453i)19-s + (0.978 + 0.207i)23-s + (0.978 − 0.207i)25-s + (−0.933 + 0.358i)29-s + (0.913 − 0.406i)31-s + (0.453 − 0.891i)35-s + (0.809 − 0.587i)37-s + (0.743 + 0.669i)43-s + (−0.998 + 0.0523i)47-s + (−0.406 − 0.913i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2952 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.931 - 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2952 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.931 - 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2952\)    =    \(2^{3} \cdot 3^{2} \cdot 41\)
Sign: $-0.931 - 0.363i$
Analytic conductor: \(317.236\)
Root analytic conductor: \(317.236\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2952} (1739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2952,\ (1:\ ),\ -0.931 - 0.363i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03938411141 + 0.2093342265i\)
\(L(\frac12)\) \(\approx\) \(-0.03938411141 + 0.2093342265i\)
\(L(1)\) \(\approx\) \(0.6832796502 + 0.1023507757i\)
\(L(1)\) \(\approx\) \(0.6832796502 + 0.1023507757i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
41 \( 1 \)
good5 \( 1 + (-0.994 + 0.104i)T \)
7 \( 1 + (-0.544 + 0.838i)T \)
11 \( 1 + (-0.629 - 0.777i)T \)
13 \( 1 + (-0.838 + 0.544i)T \)
17 \( 1 + (-0.156 - 0.987i)T \)
19 \( 1 + (-0.891 + 0.453i)T \)
23 \( 1 + (0.978 + 0.207i)T \)
29 \( 1 + (-0.933 + 0.358i)T \)
31 \( 1 + (0.913 - 0.406i)T \)
37 \( 1 + (0.809 - 0.587i)T \)
43 \( 1 + (0.743 + 0.669i)T \)
47 \( 1 + (-0.998 + 0.0523i)T \)
53 \( 1 + (-0.156 + 0.987i)T \)
59 \( 1 + (0.978 + 0.207i)T \)
61 \( 1 + (0.207 + 0.978i)T \)
67 \( 1 + (-0.629 + 0.777i)T \)
71 \( 1 + (0.987 - 0.156i)T \)
73 \( 1 + iT \)
79 \( 1 + (0.965 + 0.258i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (-0.453 - 0.891i)T \)
97 \( 1 + (0.358 + 0.933i)T \)
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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

−18.72845031718131456104484590109, −17.65558900470885353014202685015, −17.113744837667286086366725172112, −16.52236612323369741709119127577, −15.57760154379822498650997660779, −15.113205876576070366568009654088, −14.57905086956717089523097791657, −13.319153370267235999275273115014, −12.85002500622390079056508727939, −12.3763595373348956715005389281, −11.33561685259484993550803684761, −10.66274973801495378781702160738, −10.09913885291698882006166590411, −9.25655913670077091355253276256, −8.22820225065237514485909370283, −7.73323076394573115181219438325, −6.98126679432079824192136193396, −6.37819086091970386650706879350, −5.05803654007020541845383020140, −4.55130441859398274009604847621, −3.74316525767009151108505626256, −2.96261255835818845218120514589, −2.013379025119486330170931057723, −0.692288541734088714500805384087, −0.05842765243906413246933475134, 0.87407456236942340978019473449, 2.40730421837147992591186327485, 2.82328279651766248127029536741, 3.76717378406650724452564758870, 4.639379984241863539769956540915, 5.39738648220861365169533321263, 6.26834002799066695274563415143, 7.08526283819161783859597459867, 7.7651142041597177082815379062, 8.57600436235177036327153683931, 9.2133838103961456026856283114, 9.972192695298058885016868547764, 11.065160438331006261236025382540, 11.42465282111803618144376796756, 12.280013928461278972996509285506, 12.83090807663831175223916257280, 13.60778289973259893811836017650, 14.711984036874728676729883343895, 15.01370837857820127903092534526, 15.96828093451723585425569303294, 16.30267519215773458179258004881, 17.06513403608421029558280995648, 18.172121647127273840955139160674, 18.77737752554839343879886217367, 19.2265134052045554789963907750

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