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Properties

Label 2-684-171.11-c2-0-4
Degree $2$
Conductor $684$
Sign $-0.995 + 0.0993i$
Analytic cond. $18.6376$
Root an. cond. $4.31713$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.11 − 2.12i)3-s + 8.55i·5-s + (−5.68 + 9.85i)7-s + (−0.0299 + 8.99i)9-s + (12.9 + 7.47i)11-s + (−0.847 + 1.46i)13-s + (18.1 − 18.1i)15-s + (−22.9 − 13.2i)17-s + (1.22 + 18.9i)19-s + (32.9 − 8.77i)21-s + (−2.95 − 1.70i)23-s − 48.1·25-s + (19.1 − 18.9i)27-s − 11.0i·29-s + (−18.4 − 32.0i)31-s + ⋯
L(s)  = 1  + (−0.705 − 0.708i)3-s + 1.71i·5-s + (−0.812 + 1.40i)7-s + (−0.00332 + 0.999i)9-s + (1.17 + 0.679i)11-s + (−0.0651 + 0.112i)13-s + (1.21 − 1.20i)15-s + (−1.34 − 0.778i)17-s + (0.0645 + 0.997i)19-s + (1.57 − 0.417i)21-s + (−0.128 − 0.0742i)23-s − 1.92·25-s + (0.710 − 0.703i)27-s − 0.381i·29-s + (−0.596 − 1.03i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.995 + 0.0993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.995 + 0.0993i$
Analytic conductor: \(18.6376\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1),\ -0.995 + 0.0993i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6551529897\)
\(L(\frac12)\) \(\approx\) \(0.6551529897\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (2.11 + 2.12i)T \)
19 \( 1 + (-1.22 - 18.9i)T \)
good5 \( 1 - 8.55iT - 25T^{2} \)
7 \( 1 + (5.68 - 9.85i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-12.9 - 7.47i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (0.847 - 1.46i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (22.9 + 13.2i)T + (144.5 + 250. i)T^{2} \)
23 \( 1 + (2.95 + 1.70i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 11.0iT - 841T^{2} \)
31 \( 1 + (18.4 + 32.0i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 55.7T + 1.36e3T^{2} \)
41 \( 1 + 45.0iT - 1.68e3T^{2} \)
43 \( 1 + (-12.0 - 20.8i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 - 88.7iT - 2.20e3T^{2} \)
53 \( 1 + (49.9 - 28.8i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + 23.5iT - 3.48e3T^{2} \)
61 \( 1 - 32.3T + 3.72e3T^{2} \)
67 \( 1 + (28.6 - 49.5i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (-51.0 - 29.4i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-39.1 + 67.8i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-26.4 - 45.7i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (60.4 + 34.8i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (9.10 - 5.25i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (41.4 + 71.7i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−11.02233958875848355303838476632, −9.803507581724413198211322746360, −9.251101403359393170177021729868, −7.81857405832079169585248178724, −6.93684964797720775549891625086, −6.32254236454425352771124040241, −5.83477040869168151822970034863, −4.21684800059244155240566104257, −2.77307074783079440232812155297, −2.06644729399131075372242908035, 0.28642214840942271961955213983, 1.16860232508023608254520584045, 3.65475116236177422479420066017, 4.25660969023416986874299944512, 5.06120692655028805169367131658, 6.24786698931759527169881817750, 6.90613933890524553400521806055, 8.413019111243969706591108654738, 9.135080922976108863694020965850, 9.694803095025734776700560308498

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