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Properties

Label 2-29e2-29.16-c1-0-21
Degree $2$
Conductor $841$
Sign $0.791 + 0.611i$
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  + (−0.836 − 0.402i)2-s + (1.02 − 1.29i)3-s + (−0.709 − 0.889i)4-s + (0.454 + 0.218i)5-s + (−1.38 + 0.664i)6-s + (−2.58 + 3.24i)7-s + (0.648 + 2.84i)8-s + (0.0616 + 0.269i)9-s + (−0.292 − 0.366i)10-s + (−0.191 + 0.837i)11-s − 1.87·12-s + (0.822 − 3.60i)13-s + (3.47 − 1.67i)14-s + (0.750 − 0.361i)15-s + (0.0957 − 0.419i)16-s + 7.08·17-s + ⋯
L(s)  = 1  + (−0.591 − 0.284i)2-s + (0.594 − 0.744i)3-s + (−0.354 − 0.444i)4-s + (0.203 + 0.0979i)5-s + (−0.563 + 0.271i)6-s + (−0.978 + 1.22i)7-s + (0.229 + 1.00i)8-s + (0.0205 + 0.0899i)9-s + (−0.0924 − 0.115i)10-s + (−0.0576 + 0.252i)11-s − 0.541·12-s + (0.227 − 0.998i)13-s + (0.928 − 0.447i)14-s + (0.193 − 0.0933i)15-s + (0.0239 − 0.104i)16-s + 1.71·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.791 + 0.611i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.791 + 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $0.791 + 0.611i$
Analytic conductor: \(6.71541\)
Root analytic conductor: \(2.59141\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{841} (190, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 841,\ (\ :1/2),\ 0.791 + 0.611i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15154 - 0.393193i\)
\(L(\frac12)\) \(\approx\) \(1.15154 - 0.393193i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.836 + 0.402i)T + (1.24 + 1.56i)T^{2} \)
3 \( 1 + (-1.02 + 1.29i)T + (-0.667 - 2.92i)T^{2} \)
5 \( 1 + (-0.454 - 0.218i)T + (3.11 + 3.90i)T^{2} \)
7 \( 1 + (2.58 - 3.24i)T + (-1.55 - 6.82i)T^{2} \)
11 \( 1 + (0.191 - 0.837i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (-0.822 + 3.60i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 - 7.08T + 17T^{2} \)
19 \( 1 + (0.419 + 0.525i)T + (-4.22 + 18.5i)T^{2} \)
23 \( 1 + (-5.85 + 2.81i)T + (14.3 - 17.9i)T^{2} \)
31 \( 1 + (-5.90 - 2.84i)T + (19.3 + 24.2i)T^{2} \)
37 \( 1 + (-0.917 - 4.02i)T + (-33.3 + 16.0i)T^{2} \)
41 \( 1 - 0.709T + 41T^{2} \)
43 \( 1 + (-1.61 + 0.778i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (1.03 - 4.54i)T + (-42.3 - 20.3i)T^{2} \)
53 \( 1 + (7.64 + 3.68i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 - 2.21T + 59T^{2} \)
61 \( 1 + (-3.78 + 4.75i)T + (-13.5 - 59.4i)T^{2} \)
67 \( 1 + (-2.15 - 9.42i)T + (-60.3 + 29.0i)T^{2} \)
71 \( 1 + (1.52 - 6.67i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-2.26 + 1.08i)T + (45.5 - 57.0i)T^{2} \)
79 \( 1 + (-0.207 - 0.908i)T + (-71.1 + 34.2i)T^{2} \)
83 \( 1 + (-7.71 - 9.67i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-1.43 - 0.688i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 + (-6.74 - 8.45i)T + (-21.5 + 94.5i)T^{2} \)
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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

−9.975743903288906897051221104917, −9.317888858198363447879739052089, −8.368936837117226367460732139644, −7.975290218426655603020009715570, −6.67622009035521752594588387135, −5.76849199972229430758797843962, −4.99696940530293585617411228495, −3.09087344487833089394866118306, −2.42814589660155332165919243133, −1.05724858830679029663323811432, 0.934159624545047578238920335443, 3.30548817831034861463875101152, 3.65790370881775809147903268767, 4.63524424447652283341097730528, 6.15133967690348431566872989911, 7.11346507036788277436392913641, 7.77110965731041006938936917941, 8.815860696228369076717918364476, 9.596521095175223393020580420320, 9.778674492623609024059001503315

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