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Properties

Label 2-402-201.101-c1-0-15
Degree $2$
Conductor $402$
Sign $0.593 + 0.804i$
Analytic cond. $3.20998$
Root an. cond. $1.79164$
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  + (−0.0475 − 0.998i)2-s + (1.71 + 0.273i)3-s + (−0.995 + 0.0950i)4-s + (0.479 + 0.553i)5-s + (0.191 − 1.72i)6-s + (1.67 − 3.24i)7-s + (0.142 + 0.989i)8-s + (2.85 + 0.934i)9-s + (0.529 − 0.505i)10-s + (−0.608 + 1.75i)11-s + (−1.72 − 0.109i)12-s + (0.462 − 0.587i)13-s + (−3.32 − 1.51i)14-s + (0.668 + 1.07i)15-s + (0.981 − 0.189i)16-s + (−0.0617 + 0.646i)17-s + ⋯
L(s)  = 1  + (−0.0336 − 0.706i)2-s + (0.987 + 0.157i)3-s + (−0.497 + 0.0475i)4-s + (0.214 + 0.247i)5-s + (0.0782 − 0.702i)6-s + (0.632 − 1.22i)7-s + (0.0503 + 0.349i)8-s + (0.950 + 0.311i)9-s + (0.167 − 0.159i)10-s + (−0.183 + 0.529i)11-s + (−0.499 − 0.0315i)12-s + (0.128 − 0.163i)13-s + (−0.887 − 0.405i)14-s + (0.172 + 0.278i)15-s + (0.245 − 0.0473i)16-s + (−0.0149 + 0.156i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 402 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.593 + 0.804i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 402 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.593 + 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(402\)    =    \(2 \cdot 3 \cdot 67\)
Sign: $0.593 + 0.804i$
Analytic conductor: \(3.20998\)
Root analytic conductor: \(1.79164\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{402} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 402,\ (\ :1/2),\ 0.593 + 0.804i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71960 - 0.868685i\)
\(L(\frac12)\) \(\approx\) \(1.71960 - 0.868685i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0475 + 0.998i)T \)
3 \( 1 + (-1.71 - 0.273i)T \)
67 \( 1 + (6.14 + 5.40i)T \)
good5 \( 1 + (-0.479 - 0.553i)T + (-0.711 + 4.94i)T^{2} \)
7 \( 1 + (-1.67 + 3.24i)T + (-4.06 - 5.70i)T^{2} \)
11 \( 1 + (0.608 - 1.75i)T + (-8.64 - 6.79i)T^{2} \)
13 \( 1 + (-0.462 + 0.587i)T + (-3.06 - 12.6i)T^{2} \)
17 \( 1 + (0.0617 - 0.646i)T + (-16.6 - 3.21i)T^{2} \)
19 \( 1 + (0.775 - 0.399i)T + (11.0 - 15.4i)T^{2} \)
23 \( 1 + (2.73 + 0.663i)T + (20.4 + 10.5i)T^{2} \)
29 \( 1 + (-2.60 + 1.50i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.15 - 2.74i)T + (-7.30 + 30.1i)T^{2} \)
37 \( 1 + (-4.08 + 7.07i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (6.66 - 9.36i)T + (-13.4 - 38.7i)T^{2} \)
43 \( 1 + (8.71 - 3.97i)T + (28.1 - 32.4i)T^{2} \)
47 \( 1 + (0.379 - 0.398i)T + (-2.23 - 46.9i)T^{2} \)
53 \( 1 + (2.19 - 4.79i)T + (-34.7 - 40.0i)T^{2} \)
59 \( 1 + (5.38 - 0.774i)T + (56.6 - 16.6i)T^{2} \)
61 \( 1 + (3.43 - 1.18i)T + (47.9 - 37.7i)T^{2} \)
71 \( 1 + (-0.728 - 7.62i)T + (-69.7 + 13.4i)T^{2} \)
73 \( 1 + (0.0211 + 0.0611i)T + (-57.3 + 45.1i)T^{2} \)
79 \( 1 + (-4.51 - 11.2i)T + (-57.1 + 54.5i)T^{2} \)
83 \( 1 + (2.69 + 14.0i)T + (-77.0 + 30.8i)T^{2} \)
89 \( 1 + (4.02 - 13.7i)T + (-74.8 - 48.1i)T^{2} \)
97 \( 1 + (-3.37 - 1.94i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.80407702629607262056022659892, −10.27804896597400464865879918060, −9.560577000262441267265976342184, −8.309687520610362193615062104720, −7.75431035864217395238666428783, −6.58966217986462803270806295670, −4.75370411785792949234306539114, −4.04960718786183697539058374252, −2.80185088380257978260387358316, −1.50941404458900415852462532004, 1.82976407268617856048159426855, 3.22454557990892297543971868298, 4.69658689887801824823051891329, 5.67417022042558180022227481809, 6.77150231889313982146917491008, 7.961533621556382942066132591273, 8.563844274801222925349688536123, 9.182942010037939157572750741122, 10.17269018113055231130017184244, 11.56375523901063834846269639376

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