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Properties

Label 1-4235-4235.328-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.868 - 0.495i$
Analytic cond. $19.6672$
Root an. cond. $19.6672$
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  + (0.996 − 0.0855i)2-s + (−0.951 − 0.309i)3-s + (0.985 − 0.170i)4-s + (−0.974 − 0.226i)6-s + (0.967 − 0.254i)8-s + (0.809 + 0.587i)9-s + (−0.989 − 0.142i)12-s + (−0.717 − 0.696i)13-s + (0.941 − 0.336i)16-s + (0.491 − 0.870i)17-s + (0.856 + 0.516i)18-s + (−0.736 − 0.676i)19-s + (0.281 − 0.959i)23-s + (−0.998 − 0.0570i)24-s + (−0.774 − 0.633i)26-s + (−0.587 − 0.809i)27-s + ⋯
L(s)  = 1  + (0.996 − 0.0855i)2-s + (−0.951 − 0.309i)3-s + (0.985 − 0.170i)4-s + (−0.974 − 0.226i)6-s + (0.967 − 0.254i)8-s + (0.809 + 0.587i)9-s + (−0.989 − 0.142i)12-s + (−0.717 − 0.696i)13-s + (0.941 − 0.336i)16-s + (0.491 − 0.870i)17-s + (0.856 + 0.516i)18-s + (−0.736 − 0.676i)19-s + (0.281 − 0.959i)23-s + (−0.998 − 0.0570i)24-s + (−0.774 − 0.633i)26-s + (−0.587 − 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.868 - 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.868 - 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.868 - 0.495i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (328, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.868 - 0.495i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4208548801 - 1.586114139i\)
\(L(\frac12)\) \(\approx\) \(0.4208548801 - 1.586114139i\)
\(L(1)\) \(\approx\) \(1.246033611 - 0.4923066296i\)
\(L(1)\) \(\approx\) \(1.246033611 - 0.4923066296i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.996 - 0.0855i)T \)
3 \( 1 + (-0.951 - 0.309i)T \)
13 \( 1 + (-0.717 - 0.696i)T \)
17 \( 1 + (0.491 - 0.870i)T \)
19 \( 1 + (-0.736 - 0.676i)T \)
23 \( 1 + (0.281 - 0.959i)T \)
29 \( 1 + (-0.993 - 0.113i)T \)
31 \( 1 + (0.466 - 0.884i)T \)
37 \( 1 + (-0.441 - 0.897i)T \)
41 \( 1 + (0.921 - 0.389i)T \)
43 \( 1 + (0.540 + 0.841i)T \)
47 \( 1 + (-0.856 + 0.516i)T \)
53 \( 1 + (-0.336 + 0.941i)T \)
59 \( 1 + (-0.921 - 0.389i)T \)
61 \( 1 + (-0.0855 + 0.996i)T \)
67 \( 1 + (-0.755 + 0.654i)T \)
71 \( 1 + (0.198 + 0.980i)T \)
73 \( 1 + (-0.999 + 0.0285i)T \)
79 \( 1 + (0.362 + 0.931i)T \)
83 \( 1 + (-0.791 - 0.610i)T \)
89 \( 1 + (0.415 + 0.909i)T \)
97 \( 1 + (0.0570 - 0.998i)T \)
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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

−18.8291764381273612754126208824, −17.64490844353760178111257412304, −17.09838851110005905678545356122, −16.594724338593534961121535523324, −15.98001193920514176954632326972, −15.10816489991708779602021039771, −14.778593819053733869381728699493, −13.92070204314717973251674973126, −13.08025570152842788774092669942, −12.43976775649738255904733844782, −11.96954740728900187087768229997, −11.2830106028912876945236027078, −10.582055035073084255943065551646, −10.01954193685813263226084362156, −9.13097151557504885675805827472, −7.979217072924759477183807074350, −7.2838594157541577985947279726, −6.50519864204529033891844527747, −5.99517924148174977468145476333, −5.1944423452355858882295436013, −4.65075442688601711580780384461, −3.838100653591603210270792165551, −3.26088820392943002813161255559, −1.95345860530611382417204614974, −1.40040261338027245199539928431, 0.33943240051435667155758164592, 1.32587064252125971002188917076, 2.41174009674140531658744926727, 2.89803049505687581856030390643, 4.23719102397728052735781306286, 4.5898977075165264119138526124, 5.52252049752000429496274411890, 5.89739196022928455895096716694, 6.82738889111149181685119105825, 7.36402970856507743377982368542, 8.004601639723435054881358996176, 9.338211856559170183418780482431, 10.105332070584622441307179856591, 10.87640814523618815278941492877, 11.27314958806685253875913379717, 12.11988122552344469267959195760, 12.66477957151947342045407311125, 13.0946250408116063348332100843, 13.94217763896292076180457047937, 14.70399416818546634670520247016, 15.32067916538707911175737399336, 16.088248266713861069705546495827, 16.639591044384875027773696420127, 17.31801319693884128759082488854, 17.955674617640156738353654108662

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