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Properties

Label 1-2205-2205.1147-r0-0-0
Degree $1$
Conductor $2205$
Sign $0.919 - 0.393i$
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  + (−0.149 − 0.988i)2-s + (−0.955 + 0.294i)4-s + (0.433 + 0.900i)8-s + (−0.988 + 0.149i)11-s + (0.930 + 0.365i)13-s + (0.826 − 0.563i)16-s + (−0.974 + 0.222i)17-s + 19-s + (0.294 + 0.955i)22-s + (−0.294 − 0.955i)23-s + (0.222 − 0.974i)26-s + (−0.955 − 0.294i)29-s + (0.5 + 0.866i)31-s + (−0.680 − 0.733i)32-s + (0.365 + 0.930i)34-s + ⋯
L(s)  = 1  + (−0.149 − 0.988i)2-s + (−0.955 + 0.294i)4-s + (0.433 + 0.900i)8-s + (−0.988 + 0.149i)11-s + (0.930 + 0.365i)13-s + (0.826 − 0.563i)16-s + (−0.974 + 0.222i)17-s + 19-s + (0.294 + 0.955i)22-s + (−0.294 − 0.955i)23-s + (0.222 − 0.974i)26-s + (−0.955 − 0.294i)29-s + (0.5 + 0.866i)31-s + (−0.680 − 0.733i)32-s + (0.365 + 0.930i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.919 - 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.919 - 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.919 - 0.393i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (1147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ 0.919 - 0.393i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.071853742 - 0.2198297512i\)
\(L(\frac12)\) \(\approx\) \(1.071853742 - 0.2198297512i\)
\(L(1)\) \(\approx\) \(0.8029821001 - 0.3008258766i\)
\(L(1)\) \(\approx\) \(0.8029821001 - 0.3008258766i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.149 - 0.988i)T \)
11 \( 1 + (-0.988 + 0.149i)T \)
13 \( 1 + (0.930 + 0.365i)T \)
17 \( 1 + (-0.974 + 0.222i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.294 - 0.955i)T \)
29 \( 1 + (-0.955 - 0.294i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.974 - 0.222i)T \)
41 \( 1 + (-0.826 - 0.563i)T \)
43 \( 1 + (-0.563 - 0.826i)T \)
47 \( 1 + (0.149 + 0.988i)T \)
53 \( 1 + (0.974 + 0.222i)T \)
59 \( 1 + (0.0747 + 0.997i)T \)
61 \( 1 + (-0.955 - 0.294i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (-0.222 + 0.974i)T \)
73 \( 1 + (0.781 + 0.623i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (-0.930 + 0.365i)T \)
89 \( 1 + (0.623 - 0.781i)T \)
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

−19.78185356402540989368751736315, −18.640363929331676919441633743928, −18.25842745763134446886206268883, −17.73569600889021059769607048006, −16.72097224792965586111505286852, −16.17209846565471420500035073717, −15.37116016740881904118068921277, −15.10749648835252230533148730209, −13.83102407197093143491954075406, −13.42768062909455589920351783899, −12.907924369562247182166553366884, −11.62025011714687494537339185076, −10.93872050270771595792784776584, −9.96260267842361936926110218002, −9.39229442527369024890099995791, −8.42687826669914997634646646518, −7.894175530874413242088241430058, −7.14459912050351484102530837787, −6.23557374639765936861745415128, −5.54669231811372905701634217745, −4.879043278454182743416462511393, −3.87472391217172991037381482004, −3.04246761742762374561653864718, −1.72259772327853056825108766299, −0.535510781680303765316126831226, 0.80650006488165389024641912850, 1.89419680656339872572404978553, 2.617775042210851548215427844687, 3.543291642586455700214750261, 4.335663307983883614164020912502, 5.13340318465610626815971686795, 6.01202205521621781084087362356, 7.111802462078693425663937017195, 8.03482603101667054812119202376, 8.695207716942315019077122974401, 9.40289501151475734430957361715, 10.33001074649049568473453675046, 10.819192126497457533308579072126, 11.581426962450057435039178705233, 12.311977516583780049055512393046, 13.18301241048883298596949563340, 13.57559034525084014857409585462, 14.395273160650096673191485411472, 15.42301958108774560711721241200, 16.096930456586962925616760803381, 16.96952535353194978137555729329, 17.800107723009521660132662106240, 18.49640152834035221989155063669, 18.72976351914669135679804431116, 19.982885357492114401254362971

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