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Properties

Label 2-684-171.106-c1-0-18
Degree $2$
Conductor $684$
Sign $0.349 + 0.936i$
Analytic cond. $5.46176$
Root an. cond. $2.33704$
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  + (1.42 − 0.987i)3-s + 0.958·5-s + (2.11 − 3.66i)7-s + (1.05 − 2.80i)9-s + (2.66 − 4.61i)11-s + (−3.42 + 5.93i)13-s + (1.36 − 0.946i)15-s + (−2.94 + 5.10i)17-s + (−2.72 − 3.39i)19-s + (−0.606 − 7.30i)21-s + (−1.77 + 3.07i)23-s − 4.08·25-s + (−1.27 − 5.03i)27-s + 4.26·29-s + (2.25 + 3.90i)31-s + ⋯
L(s)  = 1  + (0.821 − 0.569i)3-s + 0.428·5-s + (0.800 − 1.38i)7-s + (0.350 − 0.936i)9-s + (0.802 − 1.39i)11-s + (−0.951 + 1.64i)13-s + (0.352 − 0.244i)15-s + (−0.714 + 1.23i)17-s + (−0.626 − 0.779i)19-s + (−0.132 − 1.59i)21-s + (−0.370 + 0.641i)23-s − 0.816·25-s + (−0.245 − 0.969i)27-s + 0.792·29-s + (0.404 + 0.700i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.349 + 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.349 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.349 + 0.936i$
Analytic conductor: \(5.46176\)
Root analytic conductor: \(2.33704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1/2),\ 0.349 + 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83311 - 1.27272i\)
\(L(\frac12)\) \(\approx\) \(1.83311 - 1.27272i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.42 + 0.987i)T \)
19 \( 1 + (2.72 + 3.39i)T \)
good5 \( 1 - 0.958T + 5T^{2} \)
7 \( 1 + (-2.11 + 3.66i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.66 + 4.61i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.42 - 5.93i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.94 - 5.10i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.77 - 3.07i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.26T + 29T^{2} \)
31 \( 1 + (-2.25 - 3.90i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.49T + 37T^{2} \)
41 \( 1 - 11.9T + 41T^{2} \)
43 \( 1 + (0.0297 + 0.0515i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 1.79T + 47T^{2} \)
53 \( 1 + (-0.630 - 1.09i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 3.72T + 59T^{2} \)
61 \( 1 - 3.13T + 61T^{2} \)
67 \( 1 + (5.77 - 10.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.83 + 8.38i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.40 - 5.89i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.64 + 4.58i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.39 - 4.14i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.83 - 3.17i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.46 - 5.99i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.28380331238762328498535936318, −9.240305538068527827650891696290, −8.623439138463384103445752343953, −7.70451654354914648825706942284, −6.82542263113580237392865908650, −6.18448950999933920129878272313, −4.41037359114353968674722797565, −3.86989668933076381573503198962, −2.25981475160959974321416233665, −1.20324182993487781954870648867, 2.13277784338663358768340130549, 2.62042995169871262705182630445, 4.33049375099532703772183965163, 5.02057777511425138481361588677, 6.01077464188097482150343585651, 7.43844572157525938084799863120, 8.131539082426592348483334286115, 9.036209331301322406216333971577, 9.718146028405902806073888696784, 10.31696908659761696564163218681

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