↑

Properties

Label 2-3479-497.103-c0-0-0
Degree $2$
Conductor $3479$
Sign $-0.998 + 0.0481i$
Analytic cond. $1.73624$
Root an. cond. $1.31766$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.32 + 1.22i)2-s + (0.167 + 2.24i)4-s + (−1.40 + 1.75i)8-s + (−0.988 + 0.149i)9-s + (−0.658 + 1.67i)11-s + (−1.78 + 0.268i)16-s + (−1.48 − 1.01i)18-s + (−2.92 + 1.40i)22-s + (−1.72 − 0.531i)23-s + (−0.5 − 0.866i)25-s + (1.62 + 0.781i)29-s + (−0.826 − 0.563i)32-s + (−0.500 − 2.19i)36-s + (1.32 + 1.22i)37-s + (0.0990 − 0.433i)43-s + (−3.86 − 1.19i)44-s + ⋯
L(s)  = 1  + (1.32 + 1.22i)2-s + (0.167 + 2.24i)4-s + (−1.40 + 1.75i)8-s + (−0.988 + 0.149i)9-s + (−0.658 + 1.67i)11-s + (−1.78 + 0.268i)16-s + (−1.48 − 1.01i)18-s + (−2.92 + 1.40i)22-s + (−1.72 − 0.531i)23-s + (−0.5 − 0.866i)25-s + (1.62 + 0.781i)29-s + (−0.826 − 0.563i)32-s + (−0.500 − 2.19i)36-s + (1.32 + 1.22i)37-s + (0.0990 − 0.433i)43-s + (−3.86 − 1.19i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3479 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3479 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3479\)    =    \(7^{2} \cdot 71\)
Sign: $-0.998 + 0.0481i$
Analytic conductor: \(1.73624\)
Root analytic conductor: \(1.31766\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3479} (1097, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3479,\ (\ :0),\ -0.998 + 0.0481i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.013050558\)
\(L(\frac12)\) \(\approx\) \(2.013050558\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
71 \( 1 + (0.222 - 0.974i)T \)
good2 \( 1 + (-1.32 - 1.22i)T + (0.0747 + 0.997i)T^{2} \)
3 \( 1 + (0.988 - 0.149i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.658 - 1.67i)T + (-0.733 - 0.680i)T^{2} \)
13 \( 1 + (0.222 + 0.974i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.988 + 0.149i)T^{2} \)
23 \( 1 + (1.72 + 0.531i)T + (0.826 + 0.563i)T^{2} \)
29 \( 1 + (-1.62 - 0.781i)T + (0.623 + 0.781i)T^{2} \)
31 \( 1 + (-0.955 - 0.294i)T^{2} \)
37 \( 1 + (-1.32 - 1.22i)T + (0.0747 + 0.997i)T^{2} \)
41 \( 1 + (0.222 + 0.974i)T^{2} \)
43 \( 1 + (-0.0990 + 0.433i)T + (-0.900 - 0.433i)T^{2} \)
47 \( 1 + (0.988 + 0.149i)T^{2} \)
53 \( 1 + (-1.03 - 0.702i)T + (0.365 + 0.930i)T^{2} \)
59 \( 1 + (0.733 + 0.680i)T^{2} \)
61 \( 1 + (-0.0747 - 0.997i)T^{2} \)
67 \( 1 + (0.0332 + 0.443i)T + (-0.988 + 0.149i)T^{2} \)
73 \( 1 + (-0.826 + 0.563i)T^{2} \)
79 \( 1 + (0.162 + 0.414i)T + (-0.733 + 0.680i)T^{2} \)
83 \( 1 + (0.222 - 0.974i)T^{2} \)
89 \( 1 + (0.988 + 0.149i)T^{2} \)
97 \( 1 + (0.222 - 0.974i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.676572514126730952738024839453, −8.082070127527819540437713493648, −7.57023087705085344190705627018, −6.68910537349375316815103880105, −6.14074570765453187180167650557, −5.38138694145648282346459447946, −4.60894205149171715837572392611, −4.17801913932324570305958386656, −2.93812368754134994027827920876, −2.27009646267330067009377886847, 0.71589506761087424780691175935, 2.16157322766058003208746588233, 2.89366570273437017090337065239, 3.56250124597910909089540582935, 4.30889691695803381010259690211, 5.40538673560144497929779905593, 5.82147841973316394705071767542, 6.31312648815088321440226431169, 7.80457981551701235431838455059, 8.410682897113910609760350364261

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