Properties

Degree $2$
Conductor $1734$
Sign $-0.805 - 0.591i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·2-s + (−0.707 − 0.707i)3-s − 4-s + (−1.41 − 1.41i)5-s + (−0.707 + 0.707i)6-s + i·8-s + 1.00i·9-s + (−1.41 + 1.41i)10-s + (2.82 − 2.82i)11-s + (0.707 + 0.707i)12-s + 2·13-s + 2.00i·15-s + 16-s + 1.00·18-s − 4i·19-s + (1.41 + 1.41i)20-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.408 − 0.408i)3-s − 0.5·4-s + (−0.632 − 0.632i)5-s + (−0.288 + 0.288i)6-s + 0.353i·8-s + 0.333i·9-s + (−0.447 + 0.447i)10-s + (0.852 − 0.852i)11-s + (0.204 + 0.204i)12-s + 0.554·13-s + 0.516i·15-s + 0.250·16-s + 0.235·18-s − 0.917i·19-s + (0.316 + 0.316i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.805 - 0.591i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.805 - 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1734\)    =    \(2 \cdot 3 \cdot 17^{2}\)
Sign: $-0.805 - 0.591i$
Motivic weight: \(1\)
Character: $\chi_{1734} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1734,\ (\ :1/2),\ -0.805 - 0.591i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7768861598\)
\(L(\frac12)\) \(\approx\) \(0.7768861598\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (0.707 + 0.707i)T \)
17 \( 1 \)
good5 \( 1 + (1.41 + 1.41i)T + 5iT^{2} \)
7 \( 1 - 7iT^{2} \)
11 \( 1 + (-2.82 + 2.82i)T - 11iT^{2} \)
13 \( 1 - 2T + 13T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 - 23iT^{2} \)
29 \( 1 + (7.07 + 7.07i)T + 29iT^{2} \)
31 \( 1 + (5.65 + 5.65i)T + 31iT^{2} \)
37 \( 1 + (-1.41 - 1.41i)T + 37iT^{2} \)
41 \( 1 + (-7.07 + 7.07i)T - 41iT^{2} \)
43 \( 1 - 12iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 - 12iT - 59T^{2} \)
61 \( 1 + (7.07 - 7.07i)T - 61iT^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 + 71iT^{2} \)
73 \( 1 + (-7.07 - 7.07i)T + 73iT^{2} \)
79 \( 1 + (-5.65 + 5.65i)T - 79iT^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (9.89 + 9.89i)T + 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.020592441950972739561132143517, −8.102436865409476266665150907627, −7.44527781545000573561576711323, −6.22999711527187383113308793123, −5.64567711561840772459996967512, −4.40296691061677148069820182139, −3.90869108617668724053920777837, −2.66313317371066589482155853989, −1.32624683027231639855781763160, −0.34428919154455157982916624769, 1.60428613811992023518741227950, 3.47989323980037405429060564971, 3.89336177111704867076803382650, 4.98838794897735425210161886000, 5.80467154241489835970827918340, 6.73762330629450599688885758334, 7.23587591883460382717590516392, 8.063329569668480837433520818345, 9.108853487719715451126966616775, 9.552002746007033629793805555068

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