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Properties

Label 2-29e2-29.7-c1-0-45
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0921 − 0.403i)2-s + (0.373 + 0.179i)3-s + (1.64 − 0.793i)4-s + (0.222 + 0.974i)5-s + (0.0381 − 0.167i)6-s + (−2.54 − 1.22i)7-s + (−0.988 − 1.23i)8-s + (−1.76 − 2.21i)9-s + (0.373 − 0.179i)10-s + (1.50 − 1.88i)11-s + 0.757·12-s + (1.14 − 1.42i)13-s + (−0.260 + 1.14i)14-s + (−0.0921 + 0.403i)15-s + (1.87 − 2.34i)16-s − 4.82·17-s + ⋯
L(s)  = 1  + (−0.0651 − 0.285i)2-s + (0.215 + 0.103i)3-s + (0.823 − 0.396i)4-s + (0.0995 + 0.436i)5-s + (0.0155 − 0.0682i)6-s + (−0.963 − 0.463i)7-s + (−0.349 − 0.438i)8-s + (−0.587 − 0.737i)9-s + (0.118 − 0.0568i)10-s + (0.453 − 0.569i)11-s + 0.218·12-s + (0.316 − 0.396i)13-s + (−0.0696 + 0.305i)14-s + (−0.0237 + 0.104i)15-s + (0.467 − 0.586i)16-s − 1.17·17-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: $\chi_{841} (645, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 841,\ (\ :1/2),\ -0.388 + 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.790471 - 1.19126i\)
\(L(\frac12)\) \(\approx\) \(0.790471 - 1.19126i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0921 + 0.403i)T + (-1.80 + 0.867i)T^{2} \)
3 \( 1 + (-0.373 - 0.179i)T + (1.87 + 2.34i)T^{2} \)
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 \( 1 + (-1.50 + 1.88i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (-1.14 + 1.42i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + 4.82T + 17T^{2} \)
19 \( 1 + (5.40 - 2.60i)T + (11.8 - 14.8i)T^{2} \)
23 \( 1 + (-1.70 + 7.46i)T + (-20.7 - 9.97i)T^{2} \)
31 \( 1 + (-0.905 - 3.96i)T + (-27.9 + 13.4i)T^{2} \)
37 \( 1 + (2.49 + 3.12i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 - 12.4T + 41T^{2} \)
43 \( 1 + (1.42 - 6.25i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (-3.26 + 4.09i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-1.66 - 7.29i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 - 7.65T + 59T^{2} \)
61 \( 1 + (0.746 + 0.359i)T + (38.0 + 47.6i)T^{2} \)
67 \( 1 + (3.52 + 4.42i)T + (-14.9 + 65.3i)T^{2} \)
71 \( 1 + (1.97 - 2.47i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (0.890 - 3.89i)T + (-65.7 - 31.6i)T^{2} \)
79 \( 1 + (-0.258 - 0.323i)T + (-17.5 + 77.0i)T^{2} \)
83 \( 1 + (-3.29 + 1.58i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (0.998 + 4.37i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 + (-11.2 + 5.41i)T + (60.4 - 75.8i)T^{2} \)
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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

−10.19284709423546114940794318024, −9.096913512989751684875007893018, −8.503875465621377859716626763016, −7.03022789275058784395038721335, −6.39260557239065989345460739979, −6.01458281273969769609520624925, −4.21796116255789831185436539150, −3.20712174088636956492492552182, −2.45331475648531914849479260377, −0.64211670183940325012502179823, 1.97417881063653645072917151907, 2.79757472611951062168404157234, 4.06482912104775990888371738859, 5.34381985074262594666025900630, 6.32909169944851289279432481246, 6.93365972469882923735041951961, 7.88771900841609487131243312750, 8.945592865729049632674576240469, 9.185896538751929012535218266055, 10.60331116737841236031419184864

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