Properties

Label 1-2205-2205.1424-r0-0-0
Degree $1$
Conductor $2205$
Sign $-0.391 + 0.920i$
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  + (−0.222 + 0.974i)2-s + (−0.900 − 0.433i)4-s + (0.623 − 0.781i)8-s + (0.733 + 0.680i)11-s + (−0.733 − 0.680i)13-s + (0.623 + 0.781i)16-s + (−0.826 − 0.563i)17-s + (0.5 − 0.866i)19-s + (−0.826 + 0.563i)22-s + (0.0747 + 0.997i)23-s + (0.826 − 0.563i)26-s + (−0.826 − 0.563i)29-s − 31-s + (−0.900 + 0.433i)32-s + (0.733 − 0.680i)34-s + ⋯
L(s)  = 1  + (−0.222 + 0.974i)2-s + (−0.900 − 0.433i)4-s + (0.623 − 0.781i)8-s + (0.733 + 0.680i)11-s + (−0.733 − 0.680i)13-s + (0.623 + 0.781i)16-s + (−0.826 − 0.563i)17-s + (0.5 − 0.866i)19-s + (−0.826 + 0.563i)22-s + (0.0747 + 0.997i)23-s + (0.826 − 0.563i)26-s + (−0.826 − 0.563i)29-s − 31-s + (−0.900 + 0.433i)32-s + (0.733 − 0.680i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.391 + 0.920i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (1424, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ -0.391 + 0.920i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5600005203 + 0.8470514474i\)
\(L(\frac12)\) \(\approx\) \(0.5600005203 + 0.8470514474i\)
\(L(1)\) \(\approx\) \(0.7369057454 + 0.3845652233i\)
\(L(1)\) \(\approx\) \(0.7369057454 + 0.3845652233i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.222 + 0.974i)T \)
11 \( 1 + (0.733 + 0.680i)T \)
13 \( 1 + (-0.733 - 0.680i)T \)
17 \( 1 + (-0.826 - 0.563i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.0747 + 0.997i)T \)
29 \( 1 + (-0.826 - 0.563i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.0747 + 0.997i)T \)
41 \( 1 + (0.365 + 0.930i)T \)
43 \( 1 + (-0.365 + 0.930i)T \)
47 \( 1 + (0.222 - 0.974i)T \)
53 \( 1 + (0.0747 + 0.997i)T \)
59 \( 1 + (0.623 + 0.781i)T \)
61 \( 1 + (0.900 - 0.433i)T \)
67 \( 1 - T \)
71 \( 1 + (0.900 + 0.433i)T \)
73 \( 1 + (-0.733 + 0.680i)T \)
79 \( 1 + T \)
83 \( 1 + (0.733 - 0.680i)T \)
89 \( 1 + (0.955 + 0.294i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−19.291922371074410626342582361994, −19.12404572533692077311399598279, −18.16190577983835561282248140197, −17.50622612122690755021315826236, −16.63840308969246176063382059378, −16.29607885878320383721925051639, −14.8420389714477652067563828513, −14.33733090436030360606209296240, −13.65476201719929156269264456148, −12.677901899048310664877397498499, −12.24091479482607827861665143606, −11.33601683259223727087415256680, −10.83021960963901759341070295902, −9.98614944245042087508383770928, −9.08793477058850886830698955961, −8.76239078228637753606714135259, −7.71436375909145215459044719747, −6.87715198096860867626365993928, −5.831825594184061068511320478598, −4.944025766288810120522260347278, −3.95838970769331387254014194474, −3.528660819303089711545215228797, −2.2728199259261241913754618136, −1.72563252065568599208186060851, −0.47194592690814569475524672603, 0.869590618738722776825866343851, 2.01108026044723115284385587072, 3.21174099101928364547954759748, 4.260763593130049176365247808738, 4.951496715413027291592674523958, 5.66618053253524804013539229687, 6.67023871911316934579941063238, 7.25849686609466488825532040275, 7.852228665604816608906494569527, 8.94180257686850846463196402422, 9.47249029369115457327819565890, 10.06403218670685130748840976614, 11.182514346345903703819140108214, 11.91120396401689258183087225586, 13.02572096790438663946785746441, 13.41486222470661611669018256890, 14.37948812219349218699426952546, 15.10486904902523367125170799096, 15.45038758429871963408948957900, 16.42336174606756931580225440751, 17.0955182916484056311805151553, 17.76320415072024012060862312428, 18.16748752358094563887271419659, 19.222352281829796913741569387827, 19.88775612760842532093526238217

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