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Properties

Label 1-35e2-1225.564-r0-0-0
Degree $1$
Conductor $1225$
Sign $-0.0679 + 0.997i$
Analytic cond. $5.68887$
Root an. cond. $5.68887$
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  + (0.280 + 0.959i)2-s + (−0.992 + 0.119i)3-s + (−0.842 + 0.538i)4-s + (−0.393 − 0.919i)6-s + (−0.753 − 0.657i)8-s + (0.971 − 0.237i)9-s + (0.971 + 0.237i)11-s + (0.772 − 0.635i)12-s + (−0.134 + 0.990i)13-s + (0.420 − 0.907i)16-s + (−0.887 + 0.460i)17-s + (0.5 + 0.866i)18-s + (−0.104 − 0.994i)19-s + (0.0448 + 0.998i)22-s + (0.163 − 0.986i)23-s + (0.826 + 0.563i)24-s + ⋯
L(s)  = 1  + (0.280 + 0.959i)2-s + (−0.992 + 0.119i)3-s + (−0.842 + 0.538i)4-s + (−0.393 − 0.919i)6-s + (−0.753 − 0.657i)8-s + (0.971 − 0.237i)9-s + (0.971 + 0.237i)11-s + (0.772 − 0.635i)12-s + (−0.134 + 0.990i)13-s + (0.420 − 0.907i)16-s + (−0.887 + 0.460i)17-s + (0.5 + 0.866i)18-s + (−0.104 − 0.994i)19-s + (0.0448 + 0.998i)22-s + (0.163 − 0.986i)23-s + (0.826 + 0.563i)24-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $-0.0679 + 0.997i$
Analytic conductor: \(5.68887\)
Root analytic conductor: \(5.68887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (564, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (0:\ ),\ -0.0679 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7450627971 + 0.7975000249i\)
\(L(\frac12)\) \(\approx\) \(0.7450627971 + 0.7975000249i\)
\(L(1)\) \(\approx\) \(0.7291481333 + 0.4680418933i\)
\(L(1)\) \(\approx\) \(0.7291481333 + 0.4680418933i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.280 + 0.959i)T \)
3 \( 1 + (-0.992 + 0.119i)T \)
11 \( 1 + (0.971 + 0.237i)T \)
13 \( 1 + (-0.134 + 0.990i)T \)
17 \( 1 + (-0.887 + 0.460i)T \)
19 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + (0.163 - 0.986i)T \)
29 \( 1 + (-0.963 + 0.266i)T \)
31 \( 1 + (0.913 + 0.406i)T \)
37 \( 1 + (0.772 - 0.635i)T \)
41 \( 1 + (0.858 - 0.512i)T \)
43 \( 1 + (0.222 - 0.974i)T \)
47 \( 1 + (0.999 + 0.0299i)T \)
53 \( 1 + (0.842 - 0.538i)T \)
59 \( 1 + (0.575 + 0.817i)T \)
61 \( 1 + (0.998 - 0.0598i)T \)
67 \( 1 + (-0.913 - 0.406i)T \)
71 \( 1 + (-0.0448 - 0.998i)T \)
73 \( 1 + (-0.791 + 0.611i)T \)
79 \( 1 + (0.913 - 0.406i)T \)
83 \( 1 + (-0.473 + 0.880i)T \)
89 \( 1 + (-0.925 - 0.379i)T \)
97 \( 1 + (0.809 + 0.587i)T \)
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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

−20.97500281528164595124805559573, −20.225696663565602171070028372238, −19.3847620638928594346121413504, −18.73283035034161384147747249064, −17.84323659453675564643454758087, −17.37911695030992969500463574891, −16.5031519052248369212922977863, −15.44020128879531609408003993520, −14.70612896859953335719227437616, −13.614840409719850958377363358741, −13.02898199616606924840408225443, −12.22671767190861956299702793898, −11.47420661055259486422563222679, −11.052705551307596960808185080194, −10.00538251380032481540367046234, −9.50104357457956541218045990482, −8.34162306856420298647546550320, −7.284874642110628393900804824819, −6.078354808661366357527389897430, −5.655582668051878955093560878513, −4.544996741282208207479954660976, −3.90513223029271034800227583806, −2.760480961775037997015847471403, −1.58891665551400648696741294855, −0.72708953245205973204750803069, 0.77955273876694097682455681431, 2.26955870737073519557531203473, 3.99181028281551083350762725491, 4.31755381660355322314375228777, 5.253314529235639524560315211219, 6.247120804566215452530028517061, 6.76328426954669642747005199928, 7.392273285367621334547826317993, 8.85965881049107710642058688384, 9.18835634695696167678057075297, 10.35572389678496706177802139255, 11.32248795517805796991375141449, 12.05948438033240575783058025087, 12.81277263137391689457281076150, 13.62021851017991357755080694009, 14.57047172045697668942966975495, 15.23398119419781146374479732996, 16.071368773134670730581692386483, 16.73560994159579467360068773680, 17.3308049374685564839672947778, 17.91704245490562998319486294735, 18.818241922207956849728863325717, 19.623225950666947571217713632457, 20.90854400271890189795785473592, 21.67625853200242441594231069525

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