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Properties

Label 1-35e2-1225.352-r1-0-0
Degree $1$
Conductor $1225$
Sign $0.0679 - 0.997i$
Analytic cond. $131.644$
Root an. cond. $131.644$
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  + (−0.237 + 0.971i)2-s + (−0.800 + 0.599i)3-s + (−0.887 − 0.460i)4-s + (−0.393 − 0.919i)6-s + (0.657 − 0.753i)8-s + (0.280 − 0.959i)9-s + (−0.280 − 0.959i)11-s + (0.986 − 0.163i)12-s + (−0.990 − 0.134i)13-s + (0.575 + 0.817i)16-s + (0.538 − 0.842i)17-s + (0.866 + 0.5i)18-s + (−0.913 − 0.406i)19-s + (0.998 − 0.0448i)22-s + (−0.635 + 0.772i)23-s + (−0.0747 + 0.997i)24-s + ⋯
L(s)  = 1  + (−0.237 + 0.971i)2-s + (−0.800 + 0.599i)3-s + (−0.887 − 0.460i)4-s + (−0.393 − 0.919i)6-s + (0.657 − 0.753i)8-s + (0.280 − 0.959i)9-s + (−0.280 − 0.959i)11-s + (0.986 − 0.163i)12-s + (−0.990 − 0.134i)13-s + (0.575 + 0.817i)16-s + (0.538 − 0.842i)17-s + (0.866 + 0.5i)18-s + (−0.913 − 0.406i)19-s + (0.998 − 0.0448i)22-s + (−0.635 + 0.772i)23-s + (−0.0747 + 0.997i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0679 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0679 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $0.0679 - 0.997i$
Analytic conductor: \(131.644\)
Root analytic conductor: \(131.644\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (352, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (1:\ ),\ 0.0679 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3345250557 - 0.3125293617i\)
\(L(\frac12)\) \(\approx\) \(0.3345250557 - 0.3125293617i\)
\(L(1)\) \(\approx\) \(0.5596478711 + 0.2094871826i\)
\(L(1)\) \(\approx\) \(0.5596478711 + 0.2094871826i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.237 + 0.971i)T \)
3 \( 1 + (-0.800 + 0.599i)T \)
11 \( 1 + (-0.280 - 0.959i)T \)
13 \( 1 + (-0.990 - 0.134i)T \)
17 \( 1 + (0.538 - 0.842i)T \)
19 \( 1 + (-0.913 - 0.406i)T \)
23 \( 1 + (-0.635 + 0.772i)T \)
29 \( 1 + (0.963 - 0.266i)T \)
31 \( 1 + (-0.104 - 0.994i)T \)
37 \( 1 + (0.986 - 0.163i)T \)
41 \( 1 + (0.858 - 0.512i)T \)
43 \( 1 + (0.974 + 0.222i)T \)
47 \( 1 + (0.850 + 0.525i)T \)
53 \( 1 + (0.460 - 0.887i)T \)
59 \( 1 + (-0.420 + 0.907i)T \)
61 \( 1 + (-0.447 + 0.894i)T \)
67 \( 1 + (0.994 - 0.104i)T \)
71 \( 1 + (-0.0448 - 0.998i)T \)
73 \( 1 + (-0.379 + 0.925i)T \)
79 \( 1 + (0.104 - 0.994i)T \)
83 \( 1 + (-0.880 - 0.473i)T \)
89 \( 1 + (-0.791 + 0.611i)T \)
97 \( 1 + (0.587 - 0.809i)T \)
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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

−21.32448554251679080142065427707, −20.11298882616803552361415570738, −19.629485717051215387626886279203, −18.767268061256202177399266121358, −18.20917149080342526180427694090, −17.298544945509953627579803599437, −17.02588386414913037742830862423, −15.97446938233798057103343094988, −14.6825598617728695810674304103, −14.04321908360870138713671782688, −12.79798631773521821131008189246, −12.49494797592228982503102867229, −11.986363798991657453928890753, −10.81756999400343657381711925682, −10.3584742602512209031572341833, −9.60566037387820324577346904552, −8.36652596547138786800085400302, −7.71509044490994536249523356630, −6.78832767960471906628379591856, −5.73133641132537448538860928630, −4.73243349298195193250429961114, −4.15612311901092165579420161740, −2.63475145205799452295162226128, −1.9879956761327959725030511843, −0.99072401189892794529070132198, 0.16810006956559925974506408678, 0.86123957718097262196108562730, 2.6785341783269635005489383315, 3.99589333929165702197003579208, 4.68602613084105230304701974442, 5.64647439536146497729638451291, 6.05471026334255633828548611400, 7.162727090797190113324729676491, 7.875988834334128258849188442691, 8.971343273239671862083653180800, 9.651686594803340338582139352732, 10.38054177808947694523679863564, 11.2298761911866305855935444309, 12.15607044208373929262262439825, 13.08668838151988113013544070879, 14.02260106087005362227942893088, 14.77101936053458935947847408953, 15.59628211939337858341772079858, 16.16288734534647983135199701276, 16.866724979968272937279708257636, 17.48402042048217319416115221824, 18.19564822314654485361730016049, 19.04119229157957183194106442735, 19.79561439076829026717114400852, 21.08874208101014694310004173236

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