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Properties

Label 1-4025-4025.39-r0-0-0
Degree $1$
Conductor $4025$
Sign $-0.990 + 0.139i$
Analytic cond. $18.6920$
Root an. cond. $18.6920$
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  + (−0.640 + 0.768i)2-s + (−0.861 − 0.508i)3-s + (−0.179 − 0.983i)4-s + (0.941 − 0.336i)6-s + (0.870 + 0.491i)8-s + (0.483 + 0.875i)9-s + (−0.0665 − 0.997i)11-s + (−0.345 + 0.938i)12-s + (0.254 − 0.967i)13-s + (−0.935 + 0.353i)16-s + (0.761 + 0.647i)17-s + (−0.981 − 0.189i)18-s + (0.380 + 0.924i)19-s + (0.809 + 0.587i)22-s + (−0.5 − 0.866i)24-s + ⋯
L(s)  = 1  + (−0.640 + 0.768i)2-s + (−0.861 − 0.508i)3-s + (−0.179 − 0.983i)4-s + (0.941 − 0.336i)6-s + (0.870 + 0.491i)8-s + (0.483 + 0.875i)9-s + (−0.0665 − 0.997i)11-s + (−0.345 + 0.938i)12-s + (0.254 − 0.967i)13-s + (−0.935 + 0.353i)16-s + (0.761 + 0.647i)17-s + (−0.981 − 0.189i)18-s + (0.380 + 0.924i)19-s + (0.809 + 0.587i)22-s + (−0.5 − 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4025\)    =    \(5^{2} \cdot 7 \cdot 23\)
Sign: $-0.990 + 0.139i$
Analytic conductor: \(18.6920\)
Root analytic conductor: \(18.6920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4025} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4025,\ (0:\ ),\ -0.990 + 0.139i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01028675646 + 0.1468319508i\)
\(L(\frac12)\) \(\approx\) \(0.01028675646 + 0.1468319508i\)
\(L(1)\) \(\approx\) \(0.5292979497 + 0.07734484204i\)
\(L(1)\) \(\approx\) \(0.5292979497 + 0.07734484204i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.640 + 0.768i)T \)
3 \( 1 + (-0.861 - 0.508i)T \)
11 \( 1 + (-0.0665 - 0.997i)T \)
13 \( 1 + (0.254 - 0.967i)T \)
17 \( 1 + (0.761 + 0.647i)T \)
19 \( 1 + (0.380 + 0.924i)T \)
29 \( 1 + (0.610 + 0.791i)T \)
31 \( 1 + (0.00951 + 0.999i)T \)
37 \( 1 + (-0.483 - 0.875i)T \)
41 \( 1 + (-0.921 - 0.389i)T \)
43 \( 1 + (-0.415 + 0.909i)T \)
47 \( 1 + (-0.913 + 0.406i)T \)
53 \( 1 + (-0.449 - 0.893i)T \)
59 \( 1 + (0.964 - 0.263i)T \)
61 \( 1 + (-0.398 + 0.917i)T \)
67 \( 1 + (-0.272 + 0.962i)T \)
71 \( 1 + (-0.466 + 0.884i)T \)
73 \( 1 + (-0.879 + 0.475i)T \)
79 \( 1 + (0.988 + 0.151i)T \)
83 \( 1 + (-0.974 - 0.226i)T \)
89 \( 1 + (0.953 - 0.299i)T \)
97 \( 1 + (-0.974 + 0.226i)T \)
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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

−18.19062742136622432611720528058, −17.46937817939150549975053541384, −16.928764625159356863097708769201, −16.359103445053923736341027699152, −15.60369933377288720390230468485, −14.98368871106372758323438056588, −13.80335231937963068239103501679, −13.2635806862166909465626944656, −12.19097128959465668682274729962, −11.898888810453384794496748134010, −11.32773272390835646310853471995, −10.522033263792357255296287180673, −9.76775097034990668932628067091, −9.52139251713205173738255997684, −8.64599314366648605689929526104, −7.647691663065210105734088682664, −6.96088459369824031603849799647, −6.319939943406935333212896167739, −5.0338060181497928028906678532, −4.64733463961855888844006219488, −3.813085649463296110263795831719, −2.98750373625589899901071758141, −1.97896771220222359916953929959, −1.16044297049215535807911360229, −0.06985283671728806463478499642, 1.12786678294008143475457005048, 1.50645089289101444057373427098, 2.91832447025714477684565633756, 3.88882227856425170473832900730, 5.20655159994548783479609328480, 5.43275029436038186873093761258, 6.196712800794446350076912852026, 6.80232050920016997770658715704, 7.69754378668716078430283728561, 8.20243809163377526337736119211, 8.80358737927697519910105126418, 10.12898620348593688818968137135, 10.30867859201427267658231596816, 11.0948584572431318378159212883, 11.84900709798349942456407630327, 12.71919679875957671251373395463, 13.31552065185866980938080152241, 14.20888861045232660427618874867, 14.67237174608832348739340522199, 15.82077544542437115810367545215, 16.18559430804212059939221428883, 16.7016782785486666030970407840, 17.58274791969275809497849872000, 17.91968877845123720280059786323, 18.65084840221630729925985828636

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