Properties

Label 1-435-435.122-r0-0-0
Degree $1$
Conductor $435$
Sign $0.401 - 0.916i$
Analytic cond. $2.02013$
Root an. cond. $2.02013$
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  + (0.974 − 0.222i)2-s + (0.900 − 0.433i)4-s + (0.433 − 0.900i)7-s + (0.781 − 0.623i)8-s + (0.623 − 0.781i)11-s + (−0.781 − 0.623i)13-s + (0.222 − 0.974i)14-s + (0.623 − 0.781i)16-s + i·17-s + (−0.900 + 0.433i)19-s + (0.433 − 0.900i)22-s + (−0.974 − 0.222i)23-s + (−0.900 − 0.433i)26-s − i·28-s + (0.222 + 0.974i)31-s + (0.433 − 0.900i)32-s + ⋯
L(s)  = 1  + (0.974 − 0.222i)2-s + (0.900 − 0.433i)4-s + (0.433 − 0.900i)7-s + (0.781 − 0.623i)8-s + (0.623 − 0.781i)11-s + (−0.781 − 0.623i)13-s + (0.222 − 0.974i)14-s + (0.623 − 0.781i)16-s + i·17-s + (−0.900 + 0.433i)19-s + (0.433 − 0.900i)22-s + (−0.974 − 0.222i)23-s + (−0.900 − 0.433i)26-s − i·28-s + (0.222 + 0.974i)31-s + (0.433 − 0.900i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.401 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.401 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(435\)    =    \(3 \cdot 5 \cdot 29\)
Sign: $0.401 - 0.916i$
Analytic conductor: \(2.02013\)
Root analytic conductor: \(2.02013\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{435} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 435,\ (0:\ ),\ 0.401 - 0.916i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.121084355 - 1.386863086i\)
\(L(\frac12)\) \(\approx\) \(2.121084355 - 1.386863086i\)
\(L(1)\) \(\approx\) \(1.819426442 - 0.6324687322i\)
\(L(1)\) \(\approx\) \(1.819426442 - 0.6324687322i\)

Euler product

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

−24.239767287684471239376184252040, −23.44621029741193628448681667952, −22.37329900384874243729209125158, −21.90584217273881852299105687449, −21.03453838545597957214921319706, −20.1521585649777246991221431557, −19.29336906512224941903517051614, −18.102412157030823513951155439457, −17.1739909269927317836295878222, −16.30030517732010135767425270149, −15.21639416830273904560630015698, −14.75293244175150771467936400952, −13.848226131375896736007667320273, −12.78434220455713765393922454099, −11.843375422665738018964973784175, −11.51846701363554149084510452731, −10.00467402138814839408227300578, −8.963482990005238205895988413706, −7.77172618000991118544399437088, −6.86108039328133707329223966573, −5.910129786924602029024839611441, −4.82275043543423035431388755529, −4.16682570709215566708911788590, −2.61465370487621262093878797098, −1.93231867720823595474635584861, 1.12638524296514722153365118127, 2.369123164712710301115616817761, 3.712202664236730160813041496607, 4.31671939035551166093402285467, 5.56359492307553994466060203361, 6.432597676881378134419971761786, 7.49163176157834639346264785332, 8.471933321387066201426426107221, 10.11553031535102461033821641468, 10.64896106277854118628062076158, 11.645691199050571316258068332862, 12.56756451310057292004440685661, 13.39453447724692142117351483400, 14.43129773644963398353327693958, 14.755380228164233261929742328108, 16.08917492272450968458621929540, 16.850673841267574915494409968256, 17.727460925721226805268028496784, 19.24750501343213196286292138852, 19.717345102596668590546793274976, 20.60951823387960611252307234081, 21.53109355155765308680063178263, 22.13231890409098061538396618046, 23.15947406771953183227201548585, 23.82939155398882329228574483619

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