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Properties

Label 2-29e2-29.7-c1-0-52
Degree $2$
Conductor $841$
Sign $-0.549 - 0.835i$
Analytic cond. $6.71541$
Root an. cond. $2.59141$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0990 − 0.433i)2-s + (−1.12 − 0.541i)3-s + (1.62 − 0.781i)4-s + (−0.153 − 0.674i)5-s + (−0.123 + 0.541i)6-s + (−0.321 − 0.154i)7-s + (−1.05 − 1.32i)8-s + (−0.900 − 1.12i)9-s + (−0.277 + 0.133i)10-s + (−3.07 + 3.86i)11-s − 2.24·12-s + (−3.52 + 4.41i)13-s + (−0.0353 + 0.154i)14-s + (−0.192 + 0.841i)15-s + (1.77 − 2.22i)16-s − 4.49·17-s + ⋯
L(s)  = 1  + (−0.0700 − 0.306i)2-s + (−0.648 − 0.312i)3-s + (0.811 − 0.390i)4-s + (−0.0688 − 0.301i)5-s + (−0.0504 + 0.220i)6-s + (−0.121 − 0.0585i)7-s + (−0.372 − 0.467i)8-s + (−0.300 − 0.376i)9-s + (−0.0877 + 0.0422i)10-s + (−0.928 + 1.16i)11-s − 0.648·12-s + (−0.977 + 1.22i)13-s + (−0.00944 + 0.0413i)14-s + (−0.0495 + 0.217i)15-s + (0.444 − 0.557i)16-s − 1.08·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.549 - 0.835i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.549 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $-0.549 - 0.835i$
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: \(1\)
Selberg data: \((2,\ 841,\ (\ :1/2),\ -0.549 - 0.835i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0990 + 0.433i)T + (-1.80 + 0.867i)T^{2} \)
3 \( 1 + (1.12 + 0.541i)T + (1.87 + 2.34i)T^{2} \)
5 \( 1 + (0.153 + 0.674i)T + (-4.50 + 2.16i)T^{2} \)
7 \( 1 + (0.321 + 0.154i)T + (4.36 + 5.47i)T^{2} \)
11 \( 1 + (3.07 - 3.86i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (3.52 - 4.41i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + 4.49T + 17T^{2} \)
19 \( 1 + (2.12 - 1.02i)T + (11.8 - 14.8i)T^{2} \)
23 \( 1 + (0.510 - 2.23i)T + (-20.7 - 9.97i)T^{2} \)
31 \( 1 + (1.48 + 6.52i)T + (-27.9 + 13.4i)T^{2} \)
37 \( 1 + (-3.07 - 3.86i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + 3.10T + 41T^{2} \)
43 \( 1 + (0.757 - 3.32i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (-4.01 + 5.03i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-1.04 - 4.57i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + 12.4T + 59T^{2} \)
61 \( 1 + (1.48 + 0.712i)T + (38.0 + 47.6i)T^{2} \)
67 \( 1 + (1.44 + 1.81i)T + (-14.9 + 65.3i)T^{2} \)
71 \( 1 + (4.57 - 5.73i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-1.25 + 5.48i)T + (-65.7 - 31.6i)T^{2} \)
79 \( 1 + (-2.90 - 3.64i)T + (-17.5 + 77.0i)T^{2} \)
83 \( 1 + (4.01 - 1.93i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (1.26 + 5.53i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 + (0.162 - 0.0783i)T + (60.4 - 75.8i)T^{2} \)
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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

−9.763510769814015174555959146226, −9.054522080610400386775820133609, −7.69805923740182656976211744643, −6.87129724940281186765405009508, −6.34644650382693262262677516155, −5.21381992959713315114610986057, −4.33452703496785111650132929943, −2.67023145933825030467653736504, −1.76888793264685774785890490230, 0, 2.51105190608855590858058012659, 3.11959387466050747955344128784, 4.79677992717998073763579414885, 5.60690905970408228180591704381, 6.35875962462783455554815520858, 7.33395562060815890973849566754, 8.117819848364533416821365906977, 8.839475225838035643585644853285, 10.34163496763441047826163552816

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