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Properties

Label 1-1632-1632.203-r0-0-0
Degree $1$
Conductor $1632$
Sign $-0.980 - 0.195i$
Analytic cond. $7.57897$
Root an. cond. $7.57897$
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  + (−0.707 − 0.707i)5-s − i·7-s + (−0.707 − 0.707i)11-s + (−0.707 + 0.707i)13-s + (0.707 − 0.707i)19-s + i·23-s + i·25-s + (0.707 − 0.707i)29-s + 31-s + (0.707 − 0.707i)35-s + (0.707 + 0.707i)37-s − i·41-s + (−0.707 − 0.707i)43-s − 47-s − 49-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)5-s − i·7-s + (−0.707 − 0.707i)11-s + (−0.707 + 0.707i)13-s + (0.707 − 0.707i)19-s + i·23-s + i·25-s + (0.707 − 0.707i)29-s + 31-s + (0.707 − 0.707i)35-s + (0.707 + 0.707i)37-s − i·41-s + (−0.707 − 0.707i)43-s − 47-s − 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1632 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1632 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1632\)    =    \(2^{5} \cdot 3 \cdot 17\)
Sign: $-0.980 - 0.195i$
Analytic conductor: \(7.57897\)
Root analytic conductor: \(7.57897\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1632} (203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1632,\ (0:\ ),\ -0.980 - 0.195i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01366353326 - 0.1387281813i\)
\(L(\frac12)\) \(\approx\) \(0.01366353326 - 0.1387281813i\)
\(L(1)\) \(\approx\) \(0.7308280753 - 0.03622911578i\)
\(L(1)\) \(\approx\) \(0.7308280753 - 0.03622911578i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
17 \( 1 \)
good5 \( 1 + (-0.707 - 0.707i)T \)
7 \( 1 - iT \)
11 \( 1 + (-0.707 - 0.707i)T \)
13 \( 1 + (-0.707 + 0.707i)T \)
19 \( 1 + (0.707 - 0.707i)T \)
23 \( 1 + iT \)
29 \( 1 + (0.707 - 0.707i)T \)
31 \( 1 + T \)
37 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 - iT \)
43 \( 1 + (-0.707 - 0.707i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + (-0.707 - 0.707i)T \)
61 \( 1 + (-0.707 + 0.707i)T \)
67 \( 1 + (-0.707 + 0.707i)T \)
71 \( 1 - iT \)
73 \( 1 - iT \)
79 \( 1 - T \)
83 \( 1 + (-0.707 + 0.707i)T \)
89 \( 1 - iT \)
97 \( 1 - T \)
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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

−20.566493469271061009567957314457, −19.97848008153906760657111128397, −19.49393127746801515100990207086, −18.33386276795277115437660831701, −18.040189505400037943430949810552, −17.041216908269014210193273388773, −16.253893306283077753047077496476, −15.56135857405206177877021404762, −14.684080381346888108073689400500, −14.27147994128955941258404114010, −13.20850307219523305132882730227, −12.474765319342494720001671330497, −11.73606562970311971692262531259, −10.70180886891231776244363569918, −10.31322621905196780768010018204, −9.59516300738033708631249634209, −8.06297492409492716288052671064, −7.80037627864098542094319134614, −6.98575968607704643868008146325, −6.21127432280470085465398295254, −4.87672958411070365633004196945, −4.3855161764434281157992409490, −3.218679294650149929190655531818, −2.68984844622234711504938649423, −1.26429998180309639805018878036, 0.05389464941039690990377461954, 1.39871056685447457770638908759, 2.57819431376581105506386845604, 3.31082617001539765429520742213, 4.53493760879021433949719866724, 5.11405888915113034121294975677, 5.92281627611687951769069359315, 7.002614605349410936816453328803, 7.92514334544564077473210616505, 8.51279429909524222600048664506, 9.30085105458942085422493477960, 10.01145851614948932535693940170, 11.385424830235507983142889974407, 11.66247324141355086727604043558, 12.42633188957237842518180224056, 13.3011449907065394698649622819, 13.966339110548024032313022839719, 15.138236511164887770695898565440, 15.62077309984949396174973772653, 16.19545778025689067744698263545, 17.064661683143964837571583609449, 17.85226141049278582090046620281, 18.81431453381829218505083550284, 19.265108645477176748977682543074, 19.9604061183632758985382397605

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