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Properties

Label 2-29e2-29.7-c1-0-36
Degree $2$
Conductor $841$
Sign $0.825 - 0.564i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.00933 − 0.0408i)2-s + (2.72 + 1.31i)3-s + (1.80 − 0.867i)4-s + (0.387 + 1.69i)5-s + (0.0282 − 0.123i)6-s + (−1.85 − 0.891i)7-s + (−0.104 − 0.131i)8-s + (3.84 + 4.82i)9-s + (0.0658 − 0.0317i)10-s + (1.15 − 1.44i)11-s + 6.05·12-s + (1.60 − 2.01i)13-s + (−0.0191 + 0.0839i)14-s + (−1.17 + 5.14i)15-s + (2.48 − 3.11i)16-s − 5.23·17-s + ⋯
L(s)  = 1  + (−0.00659 − 0.0289i)2-s + (1.57 + 0.758i)3-s + (0.900 − 0.433i)4-s + (0.173 + 0.759i)5-s + (0.0115 − 0.0505i)6-s + (−0.699 − 0.336i)7-s + (−0.0369 − 0.0463i)8-s + (1.28 + 1.60i)9-s + (0.0208 − 0.0100i)10-s + (0.348 − 0.436i)11-s + 1.74·12-s + (0.445 − 0.558i)13-s + (−0.00512 + 0.0224i)14-s + (−0.303 + 1.32i)15-s + (0.621 − 0.779i)16-s − 1.26·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.825 - 0.564i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.825 - 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $0.825 - 0.564i$
Analytic conductor: \(6.71541\)
Root analytic conductor: \(2.59141\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{841} (645, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 841,\ (\ :1/2),\ 0.825 - 0.564i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.96777 + 0.916993i\)
\(L(\frac12)\) \(\approx\) \(2.96777 + 0.916993i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.00933 + 0.0408i)T + (-1.80 + 0.867i)T^{2} \)
3 \( 1 + (-2.72 - 1.31i)T + (1.87 + 2.34i)T^{2} \)
5 \( 1 + (-0.387 - 1.69i)T + (-4.50 + 2.16i)T^{2} \)
7 \( 1 + (1.85 + 0.891i)T + (4.36 + 5.47i)T^{2} \)
11 \( 1 + (-1.15 + 1.44i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (-1.60 + 2.01i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + 5.23T + 17T^{2} \)
19 \( 1 + (-0.341 + 0.164i)T + (11.8 - 14.8i)T^{2} \)
23 \( 1 + (0.807 - 3.53i)T + (-20.7 - 9.97i)T^{2} \)
31 \( 1 + (-0.526 - 2.30i)T + (-27.9 + 13.4i)T^{2} \)
37 \( 1 + (-4.09 - 5.13i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + 9.44T + 41T^{2} \)
43 \( 1 + (0.221 - 0.970i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (6.33 - 7.94i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (2.20 + 9.68i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + 6.24T + 59T^{2} \)
61 \( 1 + (7.19 + 3.46i)T + (38.0 + 47.6i)T^{2} \)
67 \( 1 + (2.58 + 3.23i)T + (-14.9 + 65.3i)T^{2} \)
71 \( 1 + (1.42 - 1.78i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (2.01 - 8.83i)T + (-65.7 - 31.6i)T^{2} \)
79 \( 1 + (6.40 + 8.03i)T + (-17.5 + 77.0i)T^{2} \)
83 \( 1 + (-14.0 + 6.77i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (2.13 + 9.37i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 + (-4.66 + 2.24i)T + (60.4 - 75.8i)T^{2} \)
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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

−10.17653080986583844797728123456, −9.588921472794176030543910117508, −8.678947487919417112432067217262, −7.80910638459554744043964992640, −6.82561113943511640292322157731, −6.20801606189612670847471601640, −4.70378008539372575479448155463, −3.30172285586814824483800906434, −3.09342447631424536740946314088, −1.83988342686105969313420944315, 1.60711195565389300787116450026, 2.40670789117679634166591192913, 3.37183409862346824791727039409, 4.41720126220986427737700422564, 6.21777947415999834231903769916, 6.79108935366583505186392187739, 7.59706000969755705354750257170, 8.593574018268608896273711602769, 8.932846318030274816105971885649, 9.761506655917497165450140992474

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