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Properties

Label 2-14e2-196.103-c1-0-22
Degree $2$
Conductor $196$
Sign $-0.430 + 0.902i$
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  + (0.852 − 1.12i)2-s + (−0.330 − 0.225i)3-s + (−0.547 − 1.92i)4-s + (−0.194 − 0.0145i)5-s + (−0.535 + 0.180i)6-s + (−1.62 − 2.09i)7-s + (−2.63 − 1.02i)8-s + (−1.03 − 2.64i)9-s + (−0.181 + 0.206i)10-s + (4.10 + 1.60i)11-s + (−0.252 + 0.758i)12-s + (2.89 − 2.31i)13-s + (−3.74 + 0.0466i)14-s + (0.0608 + 0.0485i)15-s + (−3.39 + 2.10i)16-s + (4.17 + 4.50i)17-s + ⋯
L(s)  = 1  + (0.602 − 0.798i)2-s + (−0.190 − 0.130i)3-s + (−0.273 − 0.961i)4-s + (−0.0868 − 0.00650i)5-s + (−0.218 + 0.0738i)6-s + (−0.612 − 0.790i)7-s + (−0.932 − 0.360i)8-s + (−0.345 − 0.881i)9-s + (−0.0575 + 0.0653i)10-s + (1.23 + 0.485i)11-s + (−0.0727 + 0.219i)12-s + (0.804 − 0.641i)13-s + (−0.999 + 0.0124i)14-s + (0.0157 + 0.0125i)15-s + (−0.849 + 0.526i)16-s + (1.01 + 1.09i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.430 + 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.430 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $-0.430 + 0.902i$
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: \((2,\ 196,\ (\ :1/2),\ -0.430 + 0.902i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.730655 - 1.15822i\)
\(L(\frac12)\) \(\approx\) \(0.730655 - 1.15822i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.852 + 1.12i)T \)
7 \( 1 + (1.62 + 2.09i)T \)
good3 \( 1 + (0.330 + 0.225i)T + (1.09 + 2.79i)T^{2} \)
5 \( 1 + (0.194 + 0.0145i)T + (4.94 + 0.745i)T^{2} \)
11 \( 1 + (-4.10 - 1.60i)T + (8.06 + 7.48i)T^{2} \)
13 \( 1 + (-2.89 + 2.31i)T + (2.89 - 12.6i)T^{2} \)
17 \( 1 + (-4.17 - 4.50i)T + (-1.27 + 16.9i)T^{2} \)
19 \( 1 + (2.69 - 4.66i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.58 + 2.79i)T + (-1.71 - 22.9i)T^{2} \)
29 \( 1 + (1.16 + 5.11i)T + (-26.1 + 12.5i)T^{2} \)
31 \( 1 + (-4.63 - 8.03i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-10.0 - 3.09i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (1.28 + 2.66i)T + (-25.5 + 32.0i)T^{2} \)
43 \( 1 + (-0.871 + 1.81i)T + (-26.8 - 33.6i)T^{2} \)
47 \( 1 + (-0.800 + 0.120i)T + (44.9 - 13.8i)T^{2} \)
53 \( 1 + (5.82 - 1.79i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (0.925 + 12.3i)T + (-58.3 + 8.79i)T^{2} \)
61 \( 1 + (-2.27 + 7.37i)T + (-50.4 - 34.3i)T^{2} \)
67 \( 1 + (-1.40 + 0.808i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.57 + 1.04i)T + (63.9 + 30.8i)T^{2} \)
73 \( 1 + (2.28 - 15.1i)T + (-69.7 - 21.5i)T^{2} \)
79 \( 1 + (2.58 + 1.49i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.292 - 0.366i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (10.1 - 3.99i)T + (65.2 - 60.5i)T^{2} \)
97 \( 1 - 4.71iT - 97T^{2} \)
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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

−12.35151997198541639526093115626, −11.35739981024985496122271107854, −10.28899391599519325358415659328, −9.611175415252447571461743411089, −8.289403903156489036760589665777, −6.52167946866640287989799073810, −5.98479336535887875882400897261, −4.11574656073446555889938222223, −3.43273824148722965156112539099, −1.18957755622566829878345638954, 2.90583638771697462693997494596, 4.28653408680468697590565525792, 5.60920202206980605029643631210, 6.36615866425303966821139653577, 7.57816484499857025304990620835, 8.808243396499466495328308242265, 9.465662537645349459272174442348, 11.41953092538784297820965247552, 11.69020072207595928552992320390, 13.09598554744989082201461879252

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