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Properties

Label 1-1601-1601.1188-r0-0-0
Degree $1$
Conductor $1601$
Sign $0.640 - 0.768i$
Analytic cond. $7.43501$
Root an. cond. $7.43501$
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  + (0.951 + 0.309i)2-s + (−0.972 + 0.233i)3-s + (0.809 + 0.587i)4-s + (0.587 − 0.809i)5-s + (−0.996 − 0.0784i)6-s + (−0.996 + 0.0784i)7-s + (0.587 + 0.809i)8-s + (0.891 − 0.453i)9-s + (0.809 − 0.587i)10-s + (0.0784 − 0.996i)11-s + (−0.923 − 0.382i)12-s + (0.923 + 0.382i)13-s + (−0.972 − 0.233i)14-s + (−0.382 + 0.923i)15-s + (0.309 + 0.951i)16-s + (0.760 − 0.649i)17-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)2-s + (−0.972 + 0.233i)3-s + (0.809 + 0.587i)4-s + (0.587 − 0.809i)5-s + (−0.996 − 0.0784i)6-s + (−0.996 + 0.0784i)7-s + (0.587 + 0.809i)8-s + (0.891 − 0.453i)9-s + (0.809 − 0.587i)10-s + (0.0784 − 0.996i)11-s + (−0.923 − 0.382i)12-s + (0.923 + 0.382i)13-s + (−0.972 − 0.233i)14-s + (−0.382 + 0.923i)15-s + (0.309 + 0.951i)16-s + (0.760 − 0.649i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.640 - 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.640 - 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1601\)
Sign: $0.640 - 0.768i$
Analytic conductor: \(7.43501\)
Root analytic conductor: \(7.43501\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1601} (1188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1601,\ (0:\ ),\ 0.640 - 0.768i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.831231798 - 0.8573880164i\)
\(L(\frac12)\) \(\approx\) \(1.831231798 - 0.8573880164i\)
\(L(1)\) \(\approx\) \(1.443764150 - 0.04493506440i\)
\(L(1)\) \(\approx\) \(1.443764150 - 0.04493506440i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1601 \( 1 \)
good2 \( 1 + (0.951 + 0.309i)T \)
3 \( 1 + (-0.972 + 0.233i)T \)
5 \( 1 + (0.587 - 0.809i)T \)
7 \( 1 + (-0.996 + 0.0784i)T \)
11 \( 1 + (0.0784 - 0.996i)T \)
13 \( 1 + (0.923 + 0.382i)T \)
17 \( 1 + (0.760 - 0.649i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (-0.972 - 0.233i)T \)
29 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (-0.987 - 0.156i)T \)
41 \( 1 + (-0.987 - 0.156i)T \)
43 \( 1 + (-0.707 + 0.707i)T \)
47 \( 1 + (-0.987 - 0.156i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (0.0784 - 0.996i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + (0.707 - 0.707i)T \)
71 \( 1 + (-0.233 - 0.972i)T \)
73 \( 1 + (0.972 + 0.233i)T \)
79 \( 1 + T \)
83 \( 1 + (0.382 - 0.923i)T \)
89 \( 1 + (0.987 + 0.156i)T \)
97 \( 1 + (0.156 - 0.987i)T \)
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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

−20.756188833099759469084753999705, −19.91136783846765805103083366169, −19.0148247217290280941094415975, −18.42027435513121014415973589790, −17.69262057437886290986207997918, −16.712870785746616881023703262545, −16.01806701949719497056991990952, −15.354572279567450245408880527698, −14.41969392161493958788569673230, −13.697666062157668540399589914877, −12.90528110714575163411843945810, −12.36512112870093750681294868398, −11.683920298319926947074738033814, −10.55426704123473491942353688398, −10.27409600979613892001770732735, −9.66854979347298484190668864261, −7.92225586577000007144183112717, −6.838432629885477620182074830863, −6.54679713949003387018758147030, −5.715354184847585796307507137524, −5.14875152314219553369305600512, −3.75768611658722375581761154680, −3.370897586531576448346415538096, −1.99521199462946600947874062708, −1.36744400853567572256145129519, 0.60157517425390631144720546355, 1.80762275887737980942363366846, 3.15675011355697076654242581480, 3.85653393924217665922671486417, 4.88698977832554014679255797247, 5.49315030176865971701224109011, 6.30431879443134028672173777923, 6.581796350460973906277532776076, 7.898056323119112078636915669103, 8.92434716445867399098687489680, 9.74414868435040092233874839518, 10.62812716489285933305288147479, 11.57284655511879464050231020098, 12.06578396716794712916274949244, 12.93442680746437111464049737585, 13.519798337356895063424453702508, 14.06602335170289887435883131614, 15.43857378768031899815144067610, 16.12882855466806139606050655237, 16.35023646204273653730856152074, 17.036120906105012249944508282940, 17.947314707706308388344689905103, 18.806029090079822630008533204647, 19.85534364782666194464174766557, 20.68732931586905138933073114230

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