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Properties

Label 2-63e2-441.76-c0-0-0
Degree $2$
Conductor $3969$
Sign $-0.184 - 0.982i$
Analytic cond. $1.98078$
Root an. cond. $1.40740$
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  + (−0.365 − 0.930i)4-s + (−0.826 − 0.563i)7-s + (−1.94 − 0.145i)13-s + (−0.733 + 0.680i)16-s + 0.867i·19-s + (0.0747 + 0.997i)25-s + (−0.222 + 0.974i)28-s + (1.35 + 0.781i)31-s + (−0.777 − 0.974i)37-s + (−1.32 + 1.22i)43-s + (0.365 + 0.930i)49-s + (0.574 + 1.86i)52-s + (1.45 + 0.571i)61-s + (0.900 + 0.433i)64-s + (0.222 − 0.385i)67-s + ⋯
L(s)  = 1  + (−0.365 − 0.930i)4-s + (−0.826 − 0.563i)7-s + (−1.94 − 0.145i)13-s + (−0.733 + 0.680i)16-s + 0.867i·19-s + (0.0747 + 0.997i)25-s + (−0.222 + 0.974i)28-s + (1.35 + 0.781i)31-s + (−0.777 − 0.974i)37-s + (−1.32 + 1.22i)43-s + (0.365 + 0.930i)49-s + (0.574 + 1.86i)52-s + (1.45 + 0.571i)61-s + (0.900 + 0.433i)64-s + (0.222 − 0.385i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.184 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.184 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $-0.184 - 0.982i$
Analytic conductor: \(1.98078\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (2134, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ -0.184 - 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2667034299\)
\(L(\frac12)\) \(\approx\) \(0.2667034299\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (0.826 + 0.563i)T \)
good2 \( 1 + (0.365 + 0.930i)T^{2} \)
5 \( 1 + (-0.0747 - 0.997i)T^{2} \)
11 \( 1 + (0.365 + 0.930i)T^{2} \)
13 \( 1 + (1.94 + 0.145i)T + (0.988 + 0.149i)T^{2} \)
17 \( 1 + (0.222 - 0.974i)T^{2} \)
19 \( 1 - 0.867iT - T^{2} \)
23 \( 1 + (-0.733 + 0.680i)T^{2} \)
29 \( 1 + (-0.733 - 0.680i)T^{2} \)
31 \( 1 + (-1.35 - 0.781i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.777 + 0.974i)T + (-0.222 + 0.974i)T^{2} \)
41 \( 1 + (-0.0747 - 0.997i)T^{2} \)
43 \( 1 + (1.32 - 1.22i)T + (0.0747 - 0.997i)T^{2} \)
47 \( 1 + (-0.365 - 0.930i)T^{2} \)
53 \( 1 + (-0.222 - 0.974i)T^{2} \)
59 \( 1 + (-0.826 - 0.563i)T^{2} \)
61 \( 1 + (-1.45 - 0.571i)T + (0.733 + 0.680i)T^{2} \)
67 \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.222 - 0.974i)T^{2} \)
73 \( 1 + (0.678 - 1.40i)T + (-0.623 - 0.781i)T^{2} \)
79 \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.988 - 0.149i)T^{2} \)
89 \( 1 + (-0.623 - 0.781i)T^{2} \)
97 \( 1 + (0.751 - 0.433i)T + (0.5 - 0.866i)T^{2} \)
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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

−9.007752860317655221664474165333, −8.101622562568554492629749896951, −7.18971725195703012160105728802, −6.73122884875352811043791046265, −5.80630926199409185207648446036, −5.11045313197394704880434043566, −4.43773831295342678253389006657, −3.45155418083517377154631317701, −2.46960005101291716816385060249, −1.28443062214994433058593744449, 0.15124845651363764367658412117, 2.37829358801313139734161858082, 2.76691026142680667335988087841, 3.77104971322731796439120023356, 4.72179887496226611392775298024, 5.20385814836882228729965237871, 6.48045046519439984453416725389, 6.93378284922162625016327798669, 7.72317534285501382637692519186, 8.494438297271713601363213735606

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