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Properties

Label 1-1725-1725.119-r1-0-0
Degree $1$
Conductor $1725$
Sign $0.992 + 0.123i$
Analytic cond. $185.376$
Root an. cond. $185.376$
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  + (0.0855 − 0.996i)2-s + (−0.985 − 0.170i)4-s + (0.959 − 0.281i)7-s + (−0.254 + 0.967i)8-s + (−0.974 − 0.226i)11-s + (−0.610 − 0.791i)13-s + (−0.198 − 0.980i)14-s + (0.941 + 0.336i)16-s + (−0.985 + 0.170i)17-s + (0.897 − 0.441i)19-s + (−0.309 + 0.951i)22-s + (−0.841 + 0.540i)26-s + (−0.993 + 0.113i)28-s + (−0.897 − 0.441i)29-s + (−0.254 + 0.967i)31-s + (0.415 − 0.909i)32-s + ⋯
L(s)  = 1  + (0.0855 − 0.996i)2-s + (−0.985 − 0.170i)4-s + (0.959 − 0.281i)7-s + (−0.254 + 0.967i)8-s + (−0.974 − 0.226i)11-s + (−0.610 − 0.791i)13-s + (−0.198 − 0.980i)14-s + (0.941 + 0.336i)16-s + (−0.985 + 0.170i)17-s + (0.897 − 0.441i)19-s + (−0.309 + 0.951i)22-s + (−0.841 + 0.540i)26-s + (−0.993 + 0.113i)28-s + (−0.897 − 0.441i)29-s + (−0.254 + 0.967i)31-s + (0.415 − 0.909i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1725 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.992 + 0.123i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1725 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.992 + 0.123i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1725\)    =    \(3 \cdot 5^{2} \cdot 23\)
Sign: $0.992 + 0.123i$
Analytic conductor: \(185.376\)
Root analytic conductor: \(185.376\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1725} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1725,\ (1:\ ),\ 0.992 + 0.123i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8566148794 + 0.05294278382i\)
\(L(\frac12)\) \(\approx\) \(0.8566148794 + 0.05294278382i\)
\(L(1)\) \(\approx\) \(0.7581868025 - 0.4576346010i\)
\(L(1)\) \(\approx\) \(0.7581868025 - 0.4576346010i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.0855 - 0.996i)T \)
7 \( 1 + (0.959 - 0.281i)T \)
11 \( 1 + (-0.974 - 0.226i)T \)
13 \( 1 + (-0.610 - 0.791i)T \)
17 \( 1 + (-0.985 + 0.170i)T \)
19 \( 1 + (0.897 - 0.441i)T \)
29 \( 1 + (-0.897 - 0.441i)T \)
31 \( 1 + (-0.254 + 0.967i)T \)
37 \( 1 + (0.870 - 0.491i)T \)
41 \( 1 + (-0.198 + 0.980i)T \)
43 \( 1 + (-0.841 + 0.540i)T \)
47 \( 1 + (0.309 - 0.951i)T \)
53 \( 1 + (-0.0285 + 0.999i)T \)
59 \( 1 + (-0.610 - 0.791i)T \)
61 \( 1 + (-0.998 + 0.0570i)T \)
67 \( 1 + (0.921 + 0.389i)T \)
71 \( 1 + (-0.516 - 0.856i)T \)
73 \( 1 + (-0.696 - 0.717i)T \)
79 \( 1 + (-0.564 + 0.825i)T \)
83 \( 1 + (0.993 + 0.113i)T \)
89 \( 1 + (0.362 - 0.931i)T \)
97 \( 1 + (-0.993 + 0.113i)T \)
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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

−20.276852059715673547893463390842, −19.00538589234337442805395990373, −18.40024887455809908708023938992, −17.88014815609388078385533932784, −17.07189107924681402483849568118, −16.40009150423910420833985661614, −15.52738483673526286357832527189, −15.01168973737567820462779910158, −14.25519216967077971012792227474, −13.57728982759896750918597011008, −12.80464312517540777228311466989, −11.89566757045220112807266661311, −11.12716525756161621335789951230, −10.055347304041700558703427112878, −9.26841558084113407720585360009, −8.55130985121583177008328659550, −7.6245393299617151432974018087, −7.27542251938915818772616880350, −6.14686234918708524155221167652, −5.28749720021709370008569757462, −4.75134610583057794070359221480, −3.91681986467911196410697924841, −2.62110864964849980710341874916, −1.63203235465143130027498893880, −0.197547957511612048310256914102, 0.73693280416269256348036014533, 1.79438125722187754582696572430, 2.631928697952840874473114944328, 3.433904883956130643159405491492, 4.6103821988231725485200666091, 5.03296092134849564118333013151, 5.90320291005156368836201667675, 7.374921616886595751372659972627, 7.97721874782958284911338405343, 8.788547279081030820080451064174, 9.68965623630335488485105010103, 10.50278373931607412618805791815, 11.072631001315168133728725401059, 11.70358671407638749142330026096, 12.673668833195499251640896478072, 13.311834771989755886273636618, 13.93291716317002122820391756799, 14.87459060986585812131865746752, 15.41827242027272565433710374486, 16.61602403132012366660916114752, 17.49921866942006820474213748541, 18.07565296412623401904903334102, 18.5053199588216378887843799582, 19.71072775449337690305561419803, 20.12931906583591352272483050081

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