Properties

Label 2-684-76.11-c0-0-2
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

\(L(\frac{1}{2})\) \(\approx\) \(1.502031303\)
\(L(\frac12)\) \(\approx\) \(1.502031303\)
\(L(1)\) 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

−11.19237866207424476557748299959, −10.17446997074512643735909550855, −8.999310516052095031435843537619, −7.985395863959707180681766163992, −6.88730515362269825195815864517, −6.21727926806713274991732146427, −5.37185667634884437118104599082, −3.93892080431188135125227837304, −3.24101108736264869729566913663, −2.20776088424371355918668987536, 1.62463616756399609463386000202, 3.48221391099668650963820352459, 4.33196971939661838240680503940, 4.91371105985586427181296576014, 6.07691279855582409247832469859, 7.22645662142515983275379613371, 7.87673444378264271477841716317, 8.597908888672023038064110548448, 10.07615495091473951242855859389, 10.65289158505931987992786909146

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