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Properties

Label 1-2205-2205.1523-r0-0-0
Degree $1$
Conductor $2205$
Sign $-0.800 + 0.599i$
Analytic cond. $10.2399$
Root an. cond. $10.2399$
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  + (0.781 + 0.623i)2-s + (0.222 + 0.974i)4-s + (−0.433 + 0.900i)8-s + (−0.365 − 0.930i)11-s + (−0.930 + 0.365i)13-s + (−0.900 + 0.433i)16-s + (0.294 + 0.955i)17-s + (0.5 − 0.866i)19-s + (0.294 − 0.955i)22-s + (0.680 + 0.733i)23-s + (−0.955 − 0.294i)26-s + (0.955 − 0.294i)29-s + 31-s + (−0.974 − 0.222i)32-s + (−0.365 + 0.930i)34-s + ⋯
L(s)  = 1  + (0.781 + 0.623i)2-s + (0.222 + 0.974i)4-s + (−0.433 + 0.900i)8-s + (−0.365 − 0.930i)11-s + (−0.930 + 0.365i)13-s + (−0.900 + 0.433i)16-s + (0.294 + 0.955i)17-s + (0.5 − 0.866i)19-s + (0.294 − 0.955i)22-s + (0.680 + 0.733i)23-s + (−0.955 − 0.294i)26-s + (0.955 − 0.294i)29-s + 31-s + (−0.974 − 0.222i)32-s + (−0.365 + 0.930i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.800 + 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.800 + 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.800 + 0.599i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (1523, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ -0.800 + 0.599i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6219070137 + 1.868458988i\)
\(L(\frac12)\) \(\approx\) \(0.6219070137 + 1.868458988i\)
\(L(1)\) \(\approx\) \(1.227450462 + 0.7691563724i\)
\(L(1)\) \(\approx\) \(1.227450462 + 0.7691563724i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.781 + 0.623i)T \)
11 \( 1 + (-0.365 - 0.930i)T \)
13 \( 1 + (-0.930 + 0.365i)T \)
17 \( 1 + (0.294 + 0.955i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.680 + 0.733i)T \)
29 \( 1 + (0.955 - 0.294i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.680 + 0.733i)T \)
41 \( 1 + (-0.826 + 0.563i)T \)
43 \( 1 + (-0.563 + 0.826i)T \)
47 \( 1 + (0.781 + 0.623i)T \)
53 \( 1 + (0.680 + 0.733i)T \)
59 \( 1 + (-0.900 + 0.433i)T \)
61 \( 1 + (-0.222 + 0.974i)T \)
67 \( 1 - iT \)
71 \( 1 + (0.222 + 0.974i)T \)
73 \( 1 + (0.930 + 0.365i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.930 - 0.365i)T \)
89 \( 1 + (-0.988 + 0.149i)T \)
97 \( 1 + (-0.866 + 0.5i)T \)
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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

−19.63670541271863759196379047814, −18.79938647075280872573463133244, −18.22937910237395187650955111421, −17.35519904246729156060257623687, −16.45194084503768988125419963107, −15.540276850407139080136211563666, −15.0815703489307763153689701607, −14.16304311998211287049406037062, −13.76232335840523418250211745581, −12.610954597638564380187483603648, −12.297641680073948820249575622, −11.64508071241742013335547657569, −10.506461304614409572827204270683, −10.10918897155750043524260163632, −9.41083440659281201234358451368, −8.31703094893790213583451614994, −7.20977020208063176037157391950, −6.774714377654029986593313762818, −5.481702516037248336932120897834, −5.056034136186954409995505674569, −4.28292537196833429516655234902, −3.21586692599909798791259742302, −2.55966171936664274355691949361, −1.69851467348161006286785259142, −0.49178454296061556866131191237, 1.275473145736289811724422379415, 2.69538945700824460430449973043, 3.10402608319773427637982685197, 4.229206397728824234577581780560, 4.94232398445673781473740849551, 5.67350389346328453006965415288, 6.49715245839528131047969111824, 7.1981187165781149019304911164, 8.05808824440266691365917635069, 8.647508375586635254188451114, 9.62760751733335243416154942931, 10.602365336617668333860241422681, 11.495933639837652454747080252217, 12.05153868929963291501857806288, 12.89873625966322425719407280122, 13.677332349500344285521578134, 14.0505433386066302149222095352, 15.17169581792926874552278858018, 15.42764454865285753463514197308, 16.38938911310449685781624815546, 17.055981409652475864231155723290, 17.52946633352028989754945551379, 18.55577707609328352951039006294, 19.39415409940321925658088429438, 20.00414616836009087828871689634

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