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Properties

Label 2-14e2-196.111-c1-0-0
Degree $2$
Conductor $196$
Sign $-0.655 + 0.755i$
Analytic cond. $1.56506$
Root an. cond. $1.25102$
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  + (−0.865 + 1.11i)2-s + (0.0500 + 0.0627i)3-s + (−0.502 − 1.93i)4-s + (−2.66 + 2.12i)5-s + (−0.113 + 0.00166i)6-s + (−2.36 − 1.19i)7-s + (2.59 + 1.11i)8-s + (0.666 − 2.91i)9-s + (−0.0707 − 4.81i)10-s + (−1.87 + 0.427i)11-s + (0.0963 − 0.128i)12-s + (−4.45 + 1.01i)13-s + (3.37 − 1.60i)14-s + (−0.266 − 0.0608i)15-s + (−3.49 + 1.94i)16-s + (2.04 − 4.24i)17-s + ⋯
L(s)  = 1  + (−0.611 + 0.790i)2-s + (0.0289 + 0.0362i)3-s + (−0.251 − 0.967i)4-s + (−1.19 + 0.949i)5-s + (−0.0463 + 0.000681i)6-s + (−0.892 − 0.451i)7-s + (0.919 + 0.393i)8-s + (0.222 − 0.972i)9-s + (−0.0223 − 1.52i)10-s + (−0.565 + 0.128i)11-s + (0.0278 − 0.0370i)12-s + (−1.23 + 0.282i)13-s + (0.903 − 0.429i)14-s + (−0.0688 − 0.0157i)15-s + (−0.873 + 0.486i)16-s + (0.495 − 1.02i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.655 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.655 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $-0.655 + 0.755i$
Analytic conductor: \(1.56506\)
Root analytic conductor: \(1.25102\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :1/2),\ -0.655 + 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00727885 - 0.0159595i\)
\(L(\frac12)\) \(\approx\) \(0.00727885 - 0.0159595i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.865 - 1.11i)T \)
7 \( 1 + (2.36 + 1.19i)T \)
good3 \( 1 + (-0.0500 - 0.0627i)T + (-0.667 + 2.92i)T^{2} \)
5 \( 1 + (2.66 - 2.12i)T + (1.11 - 4.87i)T^{2} \)
11 \( 1 + (1.87 - 0.427i)T + (9.91 - 4.77i)T^{2} \)
13 \( 1 + (4.45 - 1.01i)T + (11.7 - 5.64i)T^{2} \)
17 \( 1 + (-2.04 + 4.24i)T + (-10.5 - 13.2i)T^{2} \)
19 \( 1 + 6.13T + 19T^{2} \)
23 \( 1 + (-3.21 - 6.67i)T + (-14.3 + 17.9i)T^{2} \)
29 \( 1 + (2.47 + 1.18i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 + 0.671T + 31T^{2} \)
37 \( 1 + (-3.19 - 1.53i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + (6.69 - 5.33i)T + (9.12 - 39.9i)T^{2} \)
43 \( 1 + (0.476 + 0.379i)T + (9.56 + 41.9i)T^{2} \)
47 \( 1 + (-2.47 - 10.8i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (4.09 - 1.97i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 + (1.77 - 2.22i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (-1.90 + 3.95i)T + (-38.0 - 47.6i)T^{2} \)
67 \( 1 + 1.25iT - 67T^{2} \)
71 \( 1 + (6.71 + 13.9i)T + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (1.18 + 0.270i)T + (65.7 + 31.6i)T^{2} \)
79 \( 1 + 13.4iT - 79T^{2} \)
83 \( 1 + (-0.443 + 1.94i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (-2.31 - 0.527i)T + (80.1 + 38.6i)T^{2} \)
97 \( 1 + 6.93iT - 97T^{2} \)
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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

−13.19893534673840617240863445343, −12.01707755170693246401220143958, −10.97992974292248802596802705621, −9.961835194373102979604331459762, −9.257659650858823386715209283692, −7.66349157708497830174704621155, −7.21134116409980749296473712053, −6.29727909654314483235943238279, −4.56107538124882188903660551923, −3.15919154678528942727685111730, 0.01771523891902453841696781937, 2.43198529959659204703060617006, 3.96697206400763362064546719133, 5.08164293848720038288756291968, 7.12084516824486849917993978155, 8.201062294413159617472435167166, 8.691464076664923270754450358682, 10.06601252708355239091999601275, 10.77636627123009328495555944137, 12.04656069743773831700796372758

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