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Properties

Label 2-14e2-196.103-c1-0-10
Degree $2$
Conductor $196$
Sign $-0.274 - 0.961i$
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  + (0.253 + 1.39i)2-s + (1.97 + 1.34i)3-s + (−1.87 + 0.705i)4-s + (−0.0159 − 0.00119i)5-s + (−1.37 + 3.09i)6-s + (2.59 − 0.509i)7-s + (−1.45 − 2.42i)8-s + (0.999 + 2.54i)9-s + (−0.00238 − 0.0224i)10-s + (−1.65 − 0.649i)11-s + (−4.65 − 1.12i)12-s + (−2.07 + 1.65i)13-s + (1.36 + 3.48i)14-s + (−0.0299 − 0.0238i)15-s + (3.00 − 2.64i)16-s + (−3.29 − 3.55i)17-s + ⋯
L(s)  = 1  + (0.179 + 0.983i)2-s + (1.14 + 0.778i)3-s + (−0.935 + 0.352i)4-s + (−0.00713 − 0.000534i)5-s + (−0.561 + 1.26i)6-s + (0.981 − 0.192i)7-s + (−0.514 − 0.857i)8-s + (0.333 + 0.849i)9-s + (−0.000753 − 0.00711i)10-s + (−0.499 − 0.195i)11-s + (−1.34 − 0.325i)12-s + (−0.574 + 0.458i)13-s + (0.365 + 0.930i)14-s + (−0.00773 − 0.00616i)15-s + (0.751 − 0.660i)16-s + (−0.799 − 0.861i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.274 - 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.274 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $-0.274 - 0.961i$
Analytic conductor: \(1.56506\)
Root analytic conductor: \(1.25102\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :1/2),\ -0.274 - 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00484 + 1.33171i\)
\(L(\frac12)\) \(\approx\) \(1.00484 + 1.33171i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.253 - 1.39i)T \)
7 \( 1 + (-2.59 + 0.509i)T \)
good3 \( 1 + (-1.97 - 1.34i)T + (1.09 + 2.79i)T^{2} \)
5 \( 1 + (0.0159 + 0.00119i)T + (4.94 + 0.745i)T^{2} \)
11 \( 1 + (1.65 + 0.649i)T + (8.06 + 7.48i)T^{2} \)
13 \( 1 + (2.07 - 1.65i)T + (2.89 - 12.6i)T^{2} \)
17 \( 1 + (3.29 + 3.55i)T + (-1.27 + 16.9i)T^{2} \)
19 \( 1 + (-0.682 + 1.18i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.75 + 4.04i)T + (-1.71 - 22.9i)T^{2} \)
29 \( 1 + (-0.487 - 2.13i)T + (-26.1 + 12.5i)T^{2} \)
31 \( 1 + (-4.96 - 8.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.78 - 0.549i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (1.78 + 3.71i)T + (-25.5 + 32.0i)T^{2} \)
43 \( 1 + (4.01 - 8.34i)T + (-26.8 - 33.6i)T^{2} \)
47 \( 1 + (-9.62 + 1.45i)T + (44.9 - 13.8i)T^{2} \)
53 \( 1 + (8.11 - 2.50i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (0.462 + 6.17i)T + (-58.3 + 8.79i)T^{2} \)
61 \( 1 + (-2.72 + 8.81i)T + (-50.4 - 34.3i)T^{2} \)
67 \( 1 + (4.78 - 2.76i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (13.9 + 3.18i)T + (63.9 + 30.8i)T^{2} \)
73 \( 1 + (1.28 - 8.51i)T + (-69.7 - 21.5i)T^{2} \)
79 \( 1 + (-0.651 - 0.375i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.16 - 2.71i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (-1.89 + 0.744i)T + (65.2 - 60.5i)T^{2} \)
97 \( 1 + 12.1iT - 97T^{2} \)
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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

−13.28750068653702685892807448560, −11.91439711636735554246167531901, −10.55377833288224807932590431915, −9.435967357886630865992349403866, −8.678630198450475656648034563126, −7.88788376360194401361318078527, −6.80775001300421689887652712568, −5.02901991567794882639610638693, −4.37083870150127012076558588862, −2.85392137985302696936546998642, 1.75886189277705367143001909470, 2.74201115539358248159595530297, 4.26780987483133871934691998856, 5.63662769703867207601917666577, 7.56786994167807595630475952083, 8.223561819348261884216899731112, 9.173737933548220510879601446389, 10.29840332968920788684471370828, 11.39675539876096614757377588380, 12.28676096263828369320244782073

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