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Properties

Label 2-29e2-29.20-c1-0-13
Degree $2$
Conductor $841$
Sign $0.999 - 0.0302i$
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  + (1.39 − 0.672i)2-s + (−1.82 − 2.29i)3-s + (0.249 − 0.312i)4-s + (0.409 − 0.197i)5-s + (−4.09 − 1.97i)6-s + (2.12 + 2.66i)7-s + (−0.551 + 2.41i)8-s + (−1.24 + 5.47i)9-s + (0.438 − 0.550i)10-s + (0.325 + 1.42i)11-s − 1.17·12-s + (0.686 + 3.00i)13-s + (4.76 + 2.29i)14-s + (−1.20 − 0.578i)15-s + (1.03 + 4.52i)16-s + 2.82·17-s + ⋯
L(s)  = 1  + (0.987 − 0.475i)2-s + (−1.05 − 1.32i)3-s + (0.124 − 0.156i)4-s + (0.183 − 0.0881i)5-s + (−1.67 − 0.805i)6-s + (0.804 + 1.00i)7-s + (−0.194 + 0.854i)8-s + (−0.416 + 1.82i)9-s + (0.138 − 0.174i)10-s + (0.0982 + 0.430i)11-s − 0.339·12-s + (0.190 + 0.834i)13-s + (1.27 + 0.613i)14-s + (−0.310 − 0.149i)15-s + (0.258 + 1.13i)16-s + 0.684·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0302i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0302i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $0.999 - 0.0302i$
Analytic conductor: \(6.71541\)
Root analytic conductor: \(2.59141\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{841} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 841,\ (\ :1/2),\ 0.999 - 0.0302i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.72455 + 0.0260492i\)
\(L(\frac12)\) \(\approx\) \(1.72455 + 0.0260492i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.39 + 0.672i)T + (1.24 - 1.56i)T^{2} \)
3 \( 1 + (1.82 + 2.29i)T + (-0.667 + 2.92i)T^{2} \)
5 \( 1 + (-0.409 + 0.197i)T + (3.11 - 3.90i)T^{2} \)
7 \( 1 + (-2.12 - 2.66i)T + (-1.55 + 6.82i)T^{2} \)
11 \( 1 + (-0.325 - 1.42i)T + (-9.91 + 4.77i)T^{2} \)
13 \( 1 + (-0.686 - 3.00i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 - 2.82T + 17T^{2} \)
19 \( 1 + (1.42 - 1.79i)T + (-4.22 - 18.5i)T^{2} \)
23 \( 1 + (0.481 + 0.231i)T + (14.3 + 17.9i)T^{2} \)
31 \( 1 + (3.23 - 1.55i)T + (19.3 - 24.2i)T^{2} \)
37 \( 1 + (-1.58 + 6.94i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 - 4.43T + 41T^{2} \)
43 \( 1 + (3.71 + 1.78i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (-0.0399 - 0.174i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (2.78 - 1.33i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 - 13.2T + 59T^{2} \)
61 \( 1 + (-1.17 - 1.47i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 + (-2.70 + 11.8i)T + (-60.3 - 29.0i)T^{2} \)
71 \( 1 + (1.00 + 4.40i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-7.45 - 3.59i)T + (45.5 + 57.0i)T^{2} \)
79 \( 1 + (3.07 - 13.4i)T + (-71.1 - 34.2i)T^{2} \)
83 \( 1 + (1.52 - 1.91i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (12.1 - 5.85i)T + (55.4 - 69.5i)T^{2} \)
97 \( 1 + (-2.93 + 3.68i)T + (-21.5 - 94.5i)T^{2} \)
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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

−10.76414311484153345452105523350, −9.304597105007218329119566622962, −8.307300225176708866927877042641, −7.53375453663342234689782917364, −6.45429643755749885002670008669, −5.55706945740527885288010525700, −5.18851761441125961147899698168, −3.94504155087509811338735011645, −2.28123141207308617224108583469, −1.64962375600069618314072416269, 0.73851097184706320931429400883, 3.38187509936278756889412195353, 4.19483100951396586779058770839, 4.83567268956933336403602609572, 5.63019999540937030921723640868, 6.23046560605011814099802014121, 7.34793676220819204084379662951, 8.500706229273867284187350494782, 9.901672420960886223718969074766, 10.10748498466387140871767406104

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