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Properties

Label 1-65e2-4225.2404-r0-0-0
Degree $1$
Conductor $4225$
Sign $-0.455 + 0.890i$
Analytic cond. $19.6208$
Root an. cond. $19.6208$
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  + (−0.989 − 0.144i)2-s + (0.527 + 0.849i)3-s + (0.958 + 0.285i)4-s + (−0.399 − 0.916i)6-s + (0.120 − 0.992i)7-s + (−0.906 − 0.421i)8-s + (−0.443 + 0.896i)9-s + (0.443 + 0.896i)11-s + (0.262 + 0.964i)12-s + (−0.262 + 0.964i)14-s + (0.836 + 0.548i)16-s + (0.906 + 0.421i)17-s + (0.568 − 0.822i)18-s + (−0.309 + 0.951i)19-s + (0.906 − 0.421i)21-s + (−0.309 − 0.951i)22-s + ⋯
L(s)  = 1  + (−0.989 − 0.144i)2-s + (0.527 + 0.849i)3-s + (0.958 + 0.285i)4-s + (−0.399 − 0.916i)6-s + (0.120 − 0.992i)7-s + (−0.906 − 0.421i)8-s + (−0.443 + 0.896i)9-s + (0.443 + 0.896i)11-s + (0.262 + 0.964i)12-s + (−0.262 + 0.964i)14-s + (0.836 + 0.548i)16-s + (0.906 + 0.421i)17-s + (0.568 − 0.822i)18-s + (−0.309 + 0.951i)19-s + (0.906 − 0.421i)21-s + (−0.309 − 0.951i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.455 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.455 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4225\)    =    \(5^{2} \cdot 13^{2}\)
Sign: $-0.455 + 0.890i$
Analytic conductor: \(19.6208\)
Root analytic conductor: \(19.6208\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4225} (2404, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4225,\ (0:\ ),\ -0.455 + 0.890i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6604224663 + 1.079287109i\)
\(L(\frac12)\) \(\approx\) \(0.6604224663 + 1.079287109i\)
\(L(1)\) \(\approx\) \(0.8173295694 + 0.3094409053i\)
\(L(1)\) \(\approx\) \(0.8173295694 + 0.3094409053i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.989 - 0.144i)T \)
3 \( 1 + (0.527 + 0.849i)T \)
7 \( 1 + (0.120 - 0.992i)T \)
11 \( 1 + (0.443 + 0.896i)T \)
17 \( 1 + (0.906 + 0.421i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
23 \( 1 + (0.809 + 0.587i)T \)
29 \( 1 + (0.715 + 0.698i)T \)
31 \( 1 + (-0.399 - 0.916i)T \)
37 \( 1 + (0.215 - 0.976i)T \)
41 \( 1 + (-0.644 + 0.764i)T \)
43 \( 1 + (0.748 + 0.663i)T \)
47 \( 1 + (-0.607 + 0.794i)T \)
53 \( 1 + (-0.981 + 0.192i)T \)
59 \( 1 + (0.262 + 0.964i)T \)
61 \( 1 + (0.485 - 0.873i)T \)
67 \( 1 + (0.958 - 0.285i)T \)
71 \( 1 + (0.527 + 0.849i)T \)
73 \( 1 + (-0.443 - 0.896i)T \)
79 \( 1 + (-0.607 + 0.794i)T \)
83 \( 1 + (-0.527 + 0.849i)T \)
89 \( 1 + (0.809 + 0.587i)T \)
97 \( 1 + (0.779 - 0.626i)T \)
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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

−18.39929128503481651486079080095, −17.464037522881149318442467879679, −17.133397349994040625851443645377, −16.09586876493702928238387881949, −15.5825629385437752635565650482, −14.704347188322117552087005997215, −14.33760914387572037431560386726, −13.39567475394024655538308666548, −12.56735174108904092614099291905, −11.83977220892060196769260554742, −11.48470659141137455453606686924, −10.5516353853708073702694363319, −9.60865077656775550437479098516, −8.92279975310003223075527490024, −8.540116904299293557214304606722, −7.9519354456728073681533900467, −6.97661557128960877006927161586, −6.54216736389083550319301942109, −5.7645133887718891388664160124, −5.00621766270164291048369278486, −3.42333726795627780079835535695, −2.86536022915552525712195836192, −2.17722918259330797626042564895, −1.27512591281814774334529948031, −0.488128193250000966048098351599, 1.14582604004877963151051050415, 1.78462758223217132823469955511, 2.8258112497972970179328929515, 3.62056307567109317118524523731, 4.15948619979550139637552125752, 5.12779749699693306646303360244, 6.12195683184097668162427547522, 7.01945730822367502557937051473, 7.75565716597661013011906025724, 8.13331801661182335065982926811, 9.134205242324034849969633707225, 9.72985018631214574403727225872, 10.132538487837286143194758483814, 10.89295123641351621760603051730, 11.385289436788445374302221157019, 12.44315545874164910888796600086, 13.01197314244843506751819018090, 14.23740902062084307438621167061, 14.57802964759182321847677786639, 15.26261685477812105463154236116, 16.11647928198879631049649219204, 16.64833239646324175406974136564, 17.171015479532221618731955287490, 17.74340351066199806950187079714, 18.74481872293793257391563029737

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