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Properties

Label 2-1734-17.4-c1-0-12
Degree $2$
Conductor $1734$
Sign $0.805 - 0.591i$
Analytic cond. $13.8460$
Root an. cond. $3.72102$
Motivic weight $1$
Arithmetic yes
Rational no
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 + (0.707 + 0.707i)6-s + (1.41 + 1.41i)7-s + i·8-s − 1.00i·9-s + (0.707 − 0.707i)12-s − 2·13-s + (1.41 − 1.41i)14-s + 16-s − 1.00·18-s − 4i·19-s − 2.00·21-s + (4.24 + 4.24i)23-s + (−0.707 − 0.707i)24-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.408 + 0.408i)3-s − 0.5·4-s + (0.288 + 0.288i)6-s + (0.534 + 0.534i)7-s + 0.353i·8-s − 0.333i·9-s + (0.204 − 0.204i)12-s − 0.554·13-s + (0.377 − 0.377i)14-s + 0.250·16-s − 0.235·18-s − 0.917i·19-s − 0.436·21-s + (0.884 + 0.884i)23-s + (−0.144 − 0.144i)24-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: $0.805 - 0.591i$
Analytic conductor: \(13.8460\)
Root analytic conductor: \(3.72102\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1734} (1483, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1734,\ (\ :1/2),\ 0.805 - 0.591i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.248567778\)
\(L(\frac12)\) \(\approx\) \(1.248567778\)
\(L(\frac{3}{2})\) 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 \( 1 - 5iT^{2} \)
7 \( 1 + (-1.41 - 1.41i)T + 7iT^{2} \)
11 \( 1 + 11iT^{2} \)
13 \( 1 + 2T + 13T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + (-4.24 - 4.24i)T + 23iT^{2} \)
29 \( 1 - 29iT^{2} \)
31 \( 1 + (-7.07 + 7.07i)T - 31iT^{2} \)
37 \( 1 + (5.65 - 5.65i)T - 37iT^{2} \)
41 \( 1 + (-4.24 - 4.24i)T + 41iT^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + 12T + 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 12iT - 59T^{2} \)
61 \( 1 + (-5.65 - 5.65i)T + 61iT^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + (4.24 - 4.24i)T - 71iT^{2} \)
73 \( 1 + (-1.41 + 1.41i)T - 73iT^{2} \)
79 \( 1 + (-7.07 - 7.07i)T + 79iT^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 - 18T + 89T^{2} \)
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

−9.502032769849529953880988152407, −8.862225519542818667493172678223, −7.959545497642377253088111400242, −7.03711775319442913621994984932, −5.95334333542040162868209689683, −5.05267202835763243327965553295, −4.58475252298858521661771854950, −3.35724658657812919504985428241, −2.46047872318857943387618632463, −1.15336931777551645630911037085, 0.57793379928984029311764867084, 1.95671592656548890617737327598, 3.41260749672248664449552372753, 4.63251421641082375167313303061, 5.08344423352849282139574048712, 6.22791142417800084409828807417, 6.79404680877315482290870501156, 7.61996036106870334690278271679, 8.236873235017306718546443915655, 9.007864099903387242451776154439

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