Properties

Label 1-435-435.428-r0-0-0
Degree $1$
Conductor $435$
Sign $0.888 + 0.458i$
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  + (0.433 + 0.900i)2-s + (−0.623 + 0.781i)4-s + (0.781 − 0.623i)7-s + (−0.974 − 0.222i)8-s + (−0.222 − 0.974i)11-s + (0.974 − 0.222i)13-s + (0.900 + 0.433i)14-s + (−0.222 − 0.974i)16-s − i·17-s + (0.623 − 0.781i)19-s + (0.781 − 0.623i)22-s + (−0.433 + 0.900i)23-s + (0.623 + 0.781i)26-s + i·28-s + (0.900 − 0.433i)31-s + (0.781 − 0.623i)32-s + ⋯
L(s)  = 1  + (0.433 + 0.900i)2-s + (−0.623 + 0.781i)4-s + (0.781 − 0.623i)7-s + (−0.974 − 0.222i)8-s + (−0.222 − 0.974i)11-s + (0.974 − 0.222i)13-s + (0.900 + 0.433i)14-s + (−0.222 − 0.974i)16-s − i·17-s + (0.623 − 0.781i)19-s + (0.781 − 0.623i)22-s + (−0.433 + 0.900i)23-s + (0.623 + 0.781i)26-s + i·28-s + (0.900 − 0.433i)31-s + (0.781 − 0.623i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(435\)    =    \(3 \cdot 5 \cdot 29\)
Sign: $0.888 + 0.458i$
Analytic conductor: \(2.02013\)
Root analytic conductor: \(2.02013\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{435} (428, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 435,\ (0:\ ),\ 0.888 + 0.458i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.603427968 + 0.3896316429i\)
\(L(\frac12)\) \(\approx\) \(1.603427968 + 0.3896316429i\)
\(L(1)\) \(\approx\) \(1.268804727 + 0.3980154308i\)
\(L(1)\) \(\approx\) \(1.268804727 + 0.3980154308i\)

Euler product

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

−23.91045636225438745621682923234, −22.98402398723837695317493569942, −22.33711813019358457782357994610, −21.15018400684017429493631425681, −20.89311991582429949837873942459, −19.90894174364455289760218355176, −18.869211850578499686719212407400, −18.17124623139197189804415587074, −17.48446573171038566246157014295, −15.99662688250105230811531312812, −15.02804208039229712564152344820, −14.37483991688466284454116127185, −13.38690624125054633363411791799, −12.356633610079866282579457815036, −11.82630076286628109673197176492, −10.72563603933062200421592029555, −10.03027181651106309284803136435, −8.82330449366758457871643248717, −8.08436534684527654932919932687, −6.466088126212388770401120840034, −5.47618976479609914492368216317, −4.54277638735999087036950836616, −3.56849036210611043626322230283, −2.21464948972390850808910397880, −1.4444826803625615931201160908, 0.94224518920578949352947088542, 2.91774584614184773221575675626, 3.91585495857107775699023229402, 4.991257967211766540834304584254, 5.81224252264012358709170221412, 6.93810170569225785855356615466, 7.84750339680569134519647634827, 8.5681378464860964802507690809, 9.67443904287456656889326037419, 11.10144453531883813507976816633, 11.70398084957535418991562412859, 13.191726944949351772354691555426, 13.7320351459407565687620401909, 14.387716363584998799441336830298, 15.69798592901610176314355286593, 16.06355138465236980401724785693, 17.217509677488854360543947824960, 17.90226723001473855902509419288, 18.68970191759861948504621918036, 20.04162164334628282678669282520, 21.014804432241386532051520335530, 21.58737088189283234207782196189, 22.759110619379027327319740101909, 23.33829019609969950113949930605, 24.31923109158561633093862129219

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