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Properties

Label 1-2205-2205.2113-r0-0-0
Degree $1$
Conductor $2205$
Sign $0.704 + 0.709i$
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  + (0.680 + 0.733i)2-s + (−0.0747 + 0.997i)4-s + (−0.781 + 0.623i)8-s + (−0.733 + 0.680i)11-s + (0.294 − 0.955i)13-s + (−0.988 − 0.149i)16-s + (0.433 − 0.900i)17-s + 19-s + (−0.997 − 0.0747i)22-s + (0.997 + 0.0747i)23-s + (0.900 − 0.433i)26-s + (−0.0747 − 0.997i)29-s + (0.5 − 0.866i)31-s + (−0.563 − 0.826i)32-s + (0.955 − 0.294i)34-s + ⋯
L(s)  = 1  + (0.680 + 0.733i)2-s + (−0.0747 + 0.997i)4-s + (−0.781 + 0.623i)8-s + (−0.733 + 0.680i)11-s + (0.294 − 0.955i)13-s + (−0.988 − 0.149i)16-s + (0.433 − 0.900i)17-s + 19-s + (−0.997 − 0.0747i)22-s + (0.997 + 0.0747i)23-s + (0.900 − 0.433i)26-s + (−0.0747 − 0.997i)29-s + (0.5 − 0.866i)31-s + (−0.563 − 0.826i)32-s + (0.955 − 0.294i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.704 + 0.709i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (2113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ 0.704 + 0.709i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.103110837 + 0.8758992633i\)
\(L(\frac12)\) \(\approx\) \(2.103110837 + 0.8758992633i\)
\(L(1)\) \(\approx\) \(1.373770327 + 0.5678627757i\)
\(L(1)\) \(\approx\) \(1.373770327 + 0.5678627757i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.680 + 0.733i)T \)
11 \( 1 + (-0.733 + 0.680i)T \)
13 \( 1 + (0.294 - 0.955i)T \)
17 \( 1 + (0.433 - 0.900i)T \)
19 \( 1 + T \)
23 \( 1 + (0.997 + 0.0747i)T \)
29 \( 1 + (-0.0747 - 0.997i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (-0.433 + 0.900i)T \)
41 \( 1 + (0.988 - 0.149i)T \)
43 \( 1 + (0.149 - 0.988i)T \)
47 \( 1 + (-0.680 - 0.733i)T \)
53 \( 1 + (-0.433 - 0.900i)T \)
59 \( 1 + (0.365 + 0.930i)T \)
61 \( 1 + (-0.0747 - 0.997i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (-0.900 + 0.433i)T \)
73 \( 1 + (0.974 + 0.222i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.294 - 0.955i)T \)
89 \( 1 + (-0.222 + 0.974i)T \)
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

−19.49591445223864232493922935863, −19.20469847201636459440598826762, −18.36574214255155633995035947038, −17.74584174751551819053661484357, −16.515982224286186603639411550161, −16.01256543919466545927056259083, −15.18339814386985025115774702587, −14.21893105144685354500951251441, −13.97288384365772813228180884045, −12.900730582656861617712983033821, −12.54176871182128363286495286102, −11.49292445273344902817898473584, −10.98279656178197759044626496286, −10.32251281255073752494861460445, −9.37163304310443855344161213203, −8.75773853320137863994078141953, −7.69283534527211020182537195512, −6.70259914329915414878242761074, −5.91872398969660046371433431305, −5.17997625944962616783239138984, −4.42053323544218984115865918841, −3.39696148756718759178113898319, −2.91920541615755658301486310651, −1.74203220835587243879862566834, −0.978477120267190571213270117488, 0.73237320504069726813537488397, 2.34098035840164723382177565461, 3.03368180198472129930845420514, 3.86349884571364244997345930661, 5.02156489175416223600156721845, 5.27439312928672515738388895596, 6.25329845662382221660112830959, 7.21366395858228589151000958965, 7.70294842071709588424094561847, 8.42789585554314437371701997826, 9.477684624207826746477275778478, 10.14841044968885332714793992718, 11.288091136168557631959703787280, 11.90232999670406089840877320089, 12.78484368111831369602486159650, 13.32745860585853205863169319609, 13.98096024339141697212649751199, 14.89784595879003088275612831984, 15.51159604630272389267698570751, 15.94748987295584030951923254418, 16.90584358386594949482885268540, 17.55206885917888733744477177986, 18.21146652533335207381601757615, 18.898437509425938421287180263890, 20.14655610393406352593968730278

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