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Properties

Label 2-29e2-29.7-c1-0-35
Degree $2$
Conductor $841$
Sign $-0.715 + 0.698i$
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  + (−0.400 − 1.75i)2-s + (0.400 + 0.193i)3-s + (−1.12 + 0.541i)4-s + (−0.0794 − 0.347i)5-s + (0.178 − 0.781i)6-s + (3.64 + 1.75i)7-s + (−0.846 − 1.06i)8-s + (−1.74 − 2.19i)9-s + (−0.579 + 0.279i)10-s + (−1.81 + 2.27i)11-s − 0.554·12-s + (3.23 − 4.05i)13-s + (1.62 − 7.11i)14-s + (0.0353 − 0.154i)15-s + (−3.07 + 3.86i)16-s − 1.10·17-s + ⋯
L(s)  = 1  + (−0.283 − 1.24i)2-s + (0.231 + 0.111i)3-s + (−0.561 + 0.270i)4-s + (−0.0355 − 0.155i)5-s + (0.0728 − 0.319i)6-s + (1.37 + 0.663i)7-s + (−0.299 − 0.375i)8-s + (−0.582 − 0.730i)9-s + (−0.183 + 0.0882i)10-s + (−0.547 + 0.686i)11-s − 0.160·12-s + (0.896 − 1.12i)13-s + (0.433 − 1.90i)14-s + (0.00912 − 0.0399i)15-s + (−0.769 + 0.965i)16-s − 0.269·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.715 + 0.698i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.715 + 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $-0.715 + 0.698i$
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: \((2,\ 841,\ (\ :1/2),\ -0.715 + 0.698i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.580281 - 1.42574i\)
\(L(\frac12)\) \(\approx\) \(0.580281 - 1.42574i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.400 + 1.75i)T + (-1.80 + 0.867i)T^{2} \)
3 \( 1 + (-0.400 - 0.193i)T + (1.87 + 2.34i)T^{2} \)
5 \( 1 + (0.0794 + 0.347i)T + (-4.50 + 2.16i)T^{2} \)
7 \( 1 + (-3.64 - 1.75i)T + (4.36 + 5.47i)T^{2} \)
11 \( 1 + (1.81 - 2.27i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (-3.23 + 4.05i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + 1.10T + 17T^{2} \)
19 \( 1 + (-1.84 + 0.888i)T + (11.8 - 14.8i)T^{2} \)
23 \( 1 + (-0.920 + 4.03i)T + (-20.7 - 9.97i)T^{2} \)
31 \( 1 + (1.41 + 6.19i)T + (-27.9 + 13.4i)T^{2} \)
37 \( 1 + (-1.81 - 2.27i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + 0.396T + 41T^{2} \)
43 \( 1 + (-1.27 + 5.59i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (-4.86 + 6.09i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-0.969 - 4.24i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + 9.10T + 59T^{2} \)
61 \( 1 + (5.44 + 2.62i)T + (38.0 + 47.6i)T^{2} \)
67 \( 1 + (-0.233 - 0.292i)T + (-14.9 + 65.3i)T^{2} \)
71 \( 1 + (7.11 - 8.91i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (1.99 - 8.72i)T + (-65.7 - 31.6i)T^{2} \)
79 \( 1 + (-0.370 - 0.464i)T + (-17.5 + 77.0i)T^{2} \)
83 \( 1 + (-8.49 + 4.09i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (-0.316 - 1.38i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 + (-14.1 + 6.83i)T + (60.4 - 75.8i)T^{2} \)
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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

−10.09879150401809113387923674923, −8.964628229845818331065009647081, −8.609254421434533063948456019987, −7.65676440017771252417203949226, −6.25528333689768553397321123557, −5.28350633128590471874320437344, −4.21463870144165965570868834597, −3.00055079315758699258559660810, −2.20979004686528551087656886064, −0.856152226246540641860134878732, 1.60633757759186216178839519752, 3.10005933893130946812700767611, 4.60007573934854440023066142610, 5.37248318541783490233565149507, 6.29526230318249301601910315016, 7.38527553669105914254601637955, 7.79058183805747283697225976868, 8.607386791077951842753632377912, 9.139994670793314227514362358255, 10.89338902313314633003840177888

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