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Properties

Label 1-73-73.28-r1-0-0
Degree $1$
Conductor $73$
Sign $-0.764 + 0.644i$
Analytic cond. $7.84493$
Root an. cond. $7.84493$
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  + (−0.939 + 0.342i)2-s + (0.866 + 0.5i)3-s + (0.766 − 0.642i)4-s + (−0.422 − 0.906i)5-s + (−0.984 − 0.173i)6-s + (−0.965 + 0.258i)7-s + (−0.5 + 0.866i)8-s + (0.5 + 0.866i)9-s + (0.707 + 0.707i)10-s + (−0.906 + 0.422i)11-s + (0.984 − 0.173i)12-s + (0.573 + 0.819i)13-s + (0.819 − 0.573i)14-s + (0.0871 − 0.996i)15-s + (0.173 − 0.984i)16-s + (−0.258 + 0.965i)17-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)2-s + (0.866 + 0.5i)3-s + (0.766 − 0.642i)4-s + (−0.422 − 0.906i)5-s + (−0.984 − 0.173i)6-s + (−0.965 + 0.258i)7-s + (−0.5 + 0.866i)8-s + (0.5 + 0.866i)9-s + (0.707 + 0.707i)10-s + (−0.906 + 0.422i)11-s + (0.984 − 0.173i)12-s + (0.573 + 0.819i)13-s + (0.819 − 0.573i)14-s + (0.0871 − 0.996i)15-s + (0.173 − 0.984i)16-s + (−0.258 + 0.965i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(73\)
Sign: $-0.764 + 0.644i$
Analytic conductor: \(7.84493\)
Root analytic conductor: \(7.84493\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{73} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 73,\ (1:\ ),\ -0.764 + 0.644i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2499930368 + 0.6843300242i\)
\(L(\frac12)\) \(\approx\) \(0.2499930368 + 0.6843300242i\)
\(L(1)\) \(\approx\) \(0.6382751887 + 0.2945884135i\)
\(L(1)\) \(\approx\) \(0.6382751887 + 0.2945884135i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad73 \( 1 \)
good2 \( 1 + (-0.939 + 0.342i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (-0.422 - 0.906i)T \)
7 \( 1 + (-0.965 + 0.258i)T \)
11 \( 1 + (-0.906 + 0.422i)T \)
13 \( 1 + (0.573 + 0.819i)T \)
17 \( 1 + (-0.258 + 0.965i)T \)
19 \( 1 + (0.342 + 0.939i)T \)
23 \( 1 + (-0.342 + 0.939i)T \)
29 \( 1 + (0.422 - 0.906i)T \)
31 \( 1 + (-0.996 + 0.0871i)T \)
37 \( 1 + (-0.939 - 0.342i)T \)
41 \( 1 + (-0.173 - 0.984i)T \)
43 \( 1 + (-0.258 - 0.965i)T \)
47 \( 1 + (0.819 + 0.573i)T \)
53 \( 1 + (0.906 + 0.422i)T \)
59 \( 1 + (-0.819 + 0.573i)T \)
61 \( 1 + (-0.984 + 0.173i)T \)
67 \( 1 + (0.984 + 0.173i)T \)
71 \( 1 + (0.939 - 0.342i)T \)
79 \( 1 + (-0.984 - 0.173i)T \)
83 \( 1 + (0.707 + 0.707i)T \)
89 \( 1 + (0.173 - 0.984i)T \)
97 \( 1 + (-0.866 + 0.5i)T \)
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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

−30.628163482315215806363066478602, −29.74598337477741500548890632110, −28.93399436611964248490758676565, −27.36281409476882277093097674561, −26.26191277939519440000396856602, −25.95250307026898576232063249103, −24.69951900841084051679138029175, −23.27724328004232937688405382140, −21.89620246772356627365151558477, −20.374048053255404913993294827938, −19.74001251226555880561681415422, −18.533735359905427915288019885347, −18.13914886487955627710558430656, −16.1313910790353872649665106283, −15.34886381542262917850284250829, −13.66032550156551798177898933519, −12.57564087603752346668849561180, −11.01812436080868614572327501391, −9.95590223411238057068857406485, −8.61437530204960071521713249018, −7.462882943041225187641788312224, −6.57077904453622257708237560573, −3.35118250680391535703325770582, −2.692644140105128203332712507276, −0.42200986340222041837335847655, 1.91789045354134420763306558149, 3.78952008572048141695355842980, 5.62445693401261183026495179911, 7.44422208370135509974782206818, 8.55408633901686565915487486856, 9.41203603707016588108920455446, 10.49722717311771499064178233019, 12.277945947821107677169165520475, 13.69225209114842040715094005941, 15.40501901660167142284048520076, 15.88564185927971804071179842522, 16.90879591364903435058970946509, 18.65340692544314190969037329533, 19.49047326621573781019343822678, 20.41719092416036415579707830476, 21.36827149528098742509492680735, 23.29335680868356887298991287345, 24.39190668599635962374750068816, 25.58278783397507341327997494539, 26.11654265654204663609671806651, 27.30585639655955379553662944444, 28.326220959166391341142766607981, 29.040337952177574583213512833888, 30.88894119962792802478418170570, 31.86213223331703110429396881653

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