↑

Properties

Label 2-680-680.37-c1-0-39
Degree $2$
Conductor $680$
Sign $0.945 - 0.326i$
Analytic cond. $5.42982$
Root an. cond. $2.33019$
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  + (−1.39 − 0.229i)2-s + (1.00 + 1.49i)3-s + (1.89 + 0.639i)4-s + (1.25 − 1.84i)5-s + (−1.05 − 2.31i)6-s + (−0.346 + 0.0689i)7-s + (−2.49 − 1.32i)8-s + (−0.0919 + 0.222i)9-s + (−2.18 + 2.28i)10-s + (−1.00 + 0.199i)11-s + (0.937 + 3.47i)12-s + 5.05i·13-s + (0.499 − 0.0167i)14-s + (4.02 + 0.0372i)15-s + (3.18 + 2.42i)16-s + (3.95 − 1.16i)17-s + ⋯
L(s)  = 1  + (−0.986 − 0.162i)2-s + (0.577 + 0.864i)3-s + (0.947 + 0.319i)4-s + (0.563 − 0.826i)5-s + (−0.429 − 0.946i)6-s + (−0.131 + 0.0260i)7-s + (−0.883 − 0.469i)8-s + (−0.0306 + 0.0740i)9-s + (−0.689 + 0.724i)10-s + (−0.302 + 0.0600i)11-s + (0.270 + 1.00i)12-s + 1.40i·13-s + (0.133 − 0.00448i)14-s + (1.03 + 0.00961i)15-s + (0.795 + 0.606i)16-s + (0.959 − 0.283i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.326i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 - 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $0.945 - 0.326i$
Analytic conductor: \(5.42982\)
Root analytic conductor: \(2.33019\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{680} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 680,\ (\ :1/2),\ 0.945 - 0.326i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33382 + 0.223988i\)
\(L(\frac12)\) \(\approx\) \(1.33382 + 0.223988i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 + 0.229i)T \)
5 \( 1 + (-1.25 + 1.84i)T \)
17 \( 1 + (-3.95 + 1.16i)T \)
good3 \( 1 + (-1.00 - 1.49i)T + (-1.14 + 2.77i)T^{2} \)
7 \( 1 + (0.346 - 0.0689i)T + (6.46 - 2.67i)T^{2} \)
11 \( 1 + (1.00 - 0.199i)T + (10.1 - 4.20i)T^{2} \)
13 \( 1 - 5.05iT - 13T^{2} \)
19 \( 1 + (-5.52 + 2.28i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (2.32 - 3.48i)T + (-8.80 - 21.2i)T^{2} \)
29 \( 1 + (-2.83 + 1.89i)T + (11.0 - 26.7i)T^{2} \)
31 \( 1 + (1.22 - 6.17i)T + (-28.6 - 11.8i)T^{2} \)
37 \( 1 + (-5.77 + 3.85i)T + (14.1 - 34.1i)T^{2} \)
41 \( 1 + (-4.58 + 6.86i)T + (-15.6 - 37.8i)T^{2} \)
43 \( 1 + (-2.48 - 5.99i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 - 4.18iT - 47T^{2} \)
53 \( 1 + (1.73 + 0.719i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (-4.26 + 10.2i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-1.42 + 2.13i)T + (-23.3 - 56.3i)T^{2} \)
67 \( 1 + (-4.31 - 4.31i)T + 67iT^{2} \)
71 \( 1 + (8.21 + 1.63i)T + (65.5 + 27.1i)T^{2} \)
73 \( 1 + (4.08 + 0.812i)T + (67.4 + 27.9i)T^{2} \)
79 \( 1 + (-2.33 + 0.465i)T + (72.9 - 30.2i)T^{2} \)
83 \( 1 + (1.68 - 4.05i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (-1.76 - 1.76i)T + 89iT^{2} \)
97 \( 1 + (17.7 + 3.53i)T + (89.6 + 37.1i)T^{2} \)
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

−10.05971573530404973585903749972, −9.470326165450717452540245153567, −9.229305535581448843063434816840, −8.232562546148888877781865766218, −7.28792226230814448144287476931, −6.15215304177852263310195150084, −4.99801456883802436173982925099, −3.82023599910812668301466488997, −2.67301321260270527302290684207, −1.25993317544595144765490366990, 1.17942284137193357727690157525, 2.50085544623729646182254368114, 3.17935388400814456695459619653, 5.49989563890686659156346262915, 6.19324037343426348988590509389, 7.29821312343151789531804946722, 7.78131981511989086421172026870, 8.442843857058795746074703696361, 9.811691054801802498659620374377, 10.13669777260724964692626746881

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