Properties

Label 2-403-403.25-c1-0-12
Degree $2$
Conductor $403$
Sign $0.996 + 0.0878i$
Analytic cond. $3.21797$
Root an. cond. $1.79387$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.94i·2-s + (1.62 + 2.82i)3-s − 1.79·4-s + (2.42 + 1.40i)5-s + (5.49 − 3.17i)6-s + (0.340 − 0.196i)7-s − 0.398i·8-s + (−3.80 + 6.59i)9-s + (2.73 − 4.73i)10-s + (−2.51 − 1.45i)11-s + (−2.92 − 5.06i)12-s + (2.70 − 2.37i)13-s + (−0.383 − 0.663i)14-s + 9.13i·15-s − 4.36·16-s + (1.09 + 1.89i)17-s + ⋯
L(s)  = 1  − 1.37i·2-s + (0.940 + 1.62i)3-s − 0.897·4-s + (1.08 + 0.627i)5-s + (2.24 − 1.29i)6-s + (0.128 − 0.0743i)7-s − 0.141i·8-s + (−1.26 + 2.19i)9-s + (0.863 − 1.49i)10-s + (−0.757 − 0.437i)11-s + (−0.844 − 1.46i)12-s + (0.751 − 0.659i)13-s + (−0.102 − 0.177i)14-s + 2.35i·15-s − 1.09·16-s + (0.264 + 0.458i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.996 + 0.0878i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.996 + 0.0878i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.08948 - 0.0919926i\)
\(L(\frac12)\) \(\approx\) \(2.08948 - 0.0919926i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-2.70 + 2.37i)T \)
31 \( 1 + (-3.52 + 4.30i)T \)
good2 \( 1 + 1.94iT - 2T^{2} \)
3 \( 1 + (-1.62 - 2.82i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-2.42 - 1.40i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.340 + 0.196i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.51 + 1.45i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.09 - 1.89i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.20 - 1.27i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 2.50T + 23T^{2} \)
29 \( 1 - 9.69T + 29T^{2} \)
37 \( 1 + (4.17 - 2.40i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.06 + 2.92i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.56 + 2.71i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 11.6iT - 47T^{2} \)
53 \( 1 + (-0.927 + 1.60i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.79 + 1.03i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 2.32T + 61T^{2} \)
67 \( 1 + (1.36 + 0.786i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.18 - 2.41i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (11.5 + 6.68i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.66 + 11.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.78 - 2.76i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 14.3iT - 89T^{2} \)
97 \( 1 - 0.850iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.68125484945877483387645226996, −10.25883643209347907323427757818, −10.07517833840149911091276272943, −8.848255260801496044940593337941, −8.161419251206062983843500190006, −6.19785996314207521686226294153, −5.01406598699361530792422760249, −3.79446484154970703973366598598, −3.01278921579453758101265271146, −2.15430495125185863530184097369, 1.55354318657746042567871496093, 2.65774283161092561292084095259, 4.86091174511386537476222829333, 6.02579681421694845221104597612, 6.61382460834767122404960560221, 7.46817529802451776024009438977, 8.472360000973392075865869595381, 8.736084483253650740558114970831, 9.911455899219352280082296131515, 11.61180973015481882132396729222

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