Properties

Label 1-435-435.353-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: $\chi_{435} (353, \cdot )$
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

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

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