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Properties

Label 2-684-76.11-c0-0-1
Degree $2$
Conductor $684$
Sign $0.977 - 0.211i$
Analytic cond. $0.341360$
Root an. cond. $0.584260$
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  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (0.707 − 1.22i)5-s + i·7-s + (−0.707 + 0.707i)8-s + (−0.366 + 1.36i)10-s + 1.41i·11-s + (0.5 + 0.866i)13-s + (−0.258 − 0.965i)14-s + (0.500 − 0.866i)16-s − i·19-s − 1.41i·20-s + (−0.366 − 1.36i)22-s + (1.22 − 0.707i)23-s + (−0.499 − 0.866i)25-s + (−0.707 − 0.707i)26-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (0.707 − 1.22i)5-s + i·7-s + (−0.707 + 0.707i)8-s + (−0.366 + 1.36i)10-s + 1.41i·11-s + (0.5 + 0.866i)13-s + (−0.258 − 0.965i)14-s + (0.500 − 0.866i)16-s − i·19-s − 1.41i·20-s + (−0.366 − 1.36i)22-s + (1.22 − 0.707i)23-s + (−0.499 − 0.866i)25-s + (−0.707 − 0.707i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.977 - 0.211i$
Analytic conductor: \(0.341360\)
Root analytic conductor: \(0.584260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :0),\ 0.977 - 0.211i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7197354055\)
\(L(\frac12)\) \(\approx\) \(0.7197354055\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 \)
19 \( 1 + iT \)
good5 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 - iT - T^{2} \)
11 \( 1 - 1.41iT - T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + iT - T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T^{2} \)
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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

−10.47085599431839238989110389440, −9.419828109934458901459713597143, −9.114883455978716337017069009793, −8.497680577964683920390266393698, −7.24610575281387056939640689881, −6.40437591139222227901002660165, −5.36869083695645184936560422916, −4.63582265453525652251748249787, −2.47272631276336195970847459109, −1.54956480626974630852139072464, 1.35869944940339697347770776127, 3.05214240173508699996308818481, 3.49667625021553192695735588764, 5.61642820980081696410362381333, 6.45535182258259207460200697164, 7.21849483497617811439403050819, 8.095347287561674444405468219115, 8.988136867121343144370165974481, 10.09200852045619827166981956165, 10.65104308463041318540122226755

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