↑

Properties

Label 2-29e2-29.25-c1-0-44
Degree $2$
Conductor $841$
Sign $-0.388 - 0.921i$
Analytic cond. $6.71541$
Root an. cond. $2.59141$
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  + (0.537 − 2.35i)2-s + (−2.17 + 1.04i)3-s + (−3.44 − 1.66i)4-s + (0.222 − 0.974i)5-s + (1.29 + 5.68i)6-s + (2.54 − 1.22i)7-s + (−2.75 + 3.45i)8-s + (1.76 − 2.21i)9-s + (−2.17 − 1.04i)10-s + (−0.258 − 0.323i)11-s + 9.24·12-s + (−2.38 − 2.99i)13-s + (−1.51 − 6.65i)14-s + (0.537 + 2.35i)15-s + (1.87 + 2.34i)16-s + 0.828·17-s + ⋯
L(s)  = 1  + (0.379 − 1.66i)2-s + (−1.25 + 0.604i)3-s + (−1.72 − 0.830i)4-s + (0.0995 − 0.436i)5-s + (0.529 + 2.31i)6-s + (0.963 − 0.463i)7-s + (−0.973 + 1.22i)8-s + (0.587 − 0.737i)9-s + (−0.687 − 0.331i)10-s + (−0.0778 − 0.0976i)11-s + 2.66·12-s + (−0.662 − 0.830i)13-s + (−0.406 − 1.77i)14-s + (0.138 + 0.607i)15-s + (0.467 + 0.586i)16-s + 0.200·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.388 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.388 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $-0.388 - 0.921i$
Analytic conductor: \(6.71541\)
Root analytic conductor: \(2.59141\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{841} (605, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 841,\ (\ :1/2),\ -0.388 - 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.335785 + 0.506040i\)
\(L(\frac12)\) \(\approx\) \(0.335785 + 0.506040i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 \)
good2 \( 1 + (-0.537 + 2.35i)T + (-1.80 - 0.867i)T^{2} \)
3 \( 1 + (2.17 - 1.04i)T + (1.87 - 2.34i)T^{2} \)
5 \( 1 + (-0.222 + 0.974i)T + (-4.50 - 2.16i)T^{2} \)
7 \( 1 + (-2.54 + 1.22i)T + (4.36 - 5.47i)T^{2} \)
11 \( 1 + (0.258 + 0.323i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (2.38 + 2.99i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 - 0.828T + 17T^{2} \)
19 \( 1 + (5.40 + 2.60i)T + (11.8 + 14.8i)T^{2} \)
23 \( 1 + (0.813 + 3.56i)T + (-20.7 + 9.97i)T^{2} \)
31 \( 1 + (2.24 - 9.81i)T + (-27.9 - 13.4i)T^{2} \)
37 \( 1 + (2.49 - 3.12i)T + (-8.23 - 36.0i)T^{2} \)
41 \( 1 + 4.48T + 41T^{2} \)
43 \( 1 + (0.797 + 3.49i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (2.02 + 2.53i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (2.11 - 9.24i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 + 3.65T + 59T^{2} \)
61 \( 1 + (-4.35 + 2.09i)T + (38.0 - 47.6i)T^{2} \)
67 \( 1 + (-3.52 + 4.42i)T + (-14.9 - 65.3i)T^{2} \)
71 \( 1 + (5.50 + 6.90i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (0.890 + 3.89i)T + (-65.7 + 31.6i)T^{2} \)
79 \( 1 + (1.50 - 1.88i)T + (-17.5 - 77.0i)T^{2} \)
83 \( 1 + (6.89 + 3.32i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (-2.77 + 12.1i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + (4.04 + 1.94i)T + (60.4 + 75.8i)T^{2} \)
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

−10.39811455563462924736929112700, −9.148713615630021827209899636287, −8.259162608920811304974537389423, −6.82072623585105240430006604506, −5.42813588529851142258175835382, −4.85718243394749850507757684544, −4.36477845089489730074203501408, −3.02672991394293779209960180779, −1.58835952916660353698575374839, −0.31125417577498656784524517216, 1.96905803319852487910584569694, 4.14169439664757818271996625007, 5.04345513545729404461244757904, 5.69965181795909166628442376419, 6.45073344265210880402976678987, 7.07005298737160746856416211202, 7.87226783169724896982865107450, 8.659131298460023053344614119594, 9.819388527114926769469642832688, 11.04439640782865660868729242728

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