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Properties

Label 2-14e2-28.11-c2-0-22
Degree $2$
Conductor $196$
Sign $0.914 + 0.405i$
Analytic cond. $5.34061$
Root an. cond. $2.31097$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.80 − 0.854i)2-s + (1.35 + 0.780i)3-s + (2.53 − 3.09i)4-s + (1.71 + 2.97i)5-s + (3.11 + 0.255i)6-s + (1.94 − 7.75i)8-s + (−3.28 − 5.68i)9-s + (5.65 + 3.91i)10-s + (7.34 + 4.24i)11-s + (5.84 − 2.19i)12-s + 18.5·13-s + 5.36i·15-s + (−3.11 − 15.6i)16-s + (−4.43 + 7.68i)17-s + (−10.7 − 7.47i)18-s + (−26.2 + 15.1i)19-s + ⋯
L(s)  = 1  + (0.904 − 0.427i)2-s + (0.450 + 0.260i)3-s + (0.634 − 0.772i)4-s + (0.343 + 0.595i)5-s + (0.518 + 0.0425i)6-s + (0.243 − 0.969i)8-s + (−0.364 − 0.631i)9-s + (0.565 + 0.391i)10-s + (0.667 + 0.385i)11-s + (0.486 − 0.183i)12-s + 1.42·13-s + 0.357i·15-s + (−0.194 − 0.980i)16-s + (−0.261 + 0.452i)17-s + (−0.599 − 0.415i)18-s + (−1.38 + 0.798i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.405i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.914 + 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $0.914 + 0.405i$
Analytic conductor: \(5.34061\)
Root analytic conductor: \(2.31097\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :1),\ 0.914 + 0.405i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.97544 - 0.630069i\)
\(L(\frac12)\) \(\approx\) \(2.97544 - 0.630069i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.80 + 0.854i)T \)
7 \( 1 \)
good3 \( 1 + (-1.35 - 0.780i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (-1.71 - 2.97i)T + (-12.5 + 21.6i)T^{2} \)
11 \( 1 + (-7.34 - 4.24i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 - 18.5T + 169T^{2} \)
17 \( 1 + (4.43 - 7.68i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (26.2 - 15.1i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (22.9 - 13.2i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 18.6T + 841T^{2} \)
31 \( 1 + (35.7 + 20.6i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-1.74 - 3.02i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 37.7T + 1.68e3T^{2} \)
43 \( 1 - 50.8iT - 1.84e3T^{2} \)
47 \( 1 + (45.0 - 25.9i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (7.69 - 13.3i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-33.1 - 19.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (36.2 + 62.8i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-27.7 - 16.0i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 50.6iT - 5.04e3T^{2} \)
73 \( 1 + (2.74 - 4.75i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-34.5 + 19.9i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 4.28iT - 6.88e3T^{2} \)
89 \( 1 + (-61.9 - 107. i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 32.0T + 9.40e3T^{2} \)
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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

−12.26155273922073074268695943335, −11.26935342191250989733419556760, −10.41331971080300482001782932830, −9.458878064216758747486261428763, −8.262168957899076911919019228997, −6.45485199456631634373776591592, −6.09761708479514882941719580159, −4.20211331419642424437116655684, −3.39830541055806213066296113559, −1.85953888828360039959162501390, 1.98756610177178629955229592967, 3.54803300798335900030899205303, 4.84922237305552748274417419146, 6.00238242178717445350247021972, 6.98129610052626382982438232642, 8.528570596973635587422371844526, 8.707769572644930136250563394840, 10.68988622156330014999168795339, 11.50971819295901317777567417833, 12.71084725477574648874867889330

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