Properties

Label 2-21e2-9.7-c1-0-6
Degree $2$
Conductor $441$
Sign $0.722 - 0.690i$
Analytic cond. $3.52140$
Root an. cond. $1.87654$
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  + (−0.0341 + 0.0592i)2-s + (−1.69 + 0.355i)3-s + (0.997 + 1.72i)4-s + (−1.33 − 2.30i)5-s + (0.0368 − 0.112i)6-s − 0.273·8-s + (2.74 − 1.20i)9-s + 0.182·10-s + (0.799 − 1.38i)11-s + (−2.30 − 2.57i)12-s + (2.62 + 4.54i)13-s + (3.07 + 3.43i)15-s + (−1.98 + 3.43i)16-s + 6.54·17-s + (−0.0225 + 0.203i)18-s − 1.90·19-s + ⋯
L(s)  = 1  + (−0.0241 + 0.0418i)2-s + (−0.978 + 0.205i)3-s + (0.498 + 0.864i)4-s + (−0.595 − 1.03i)5-s + (0.0150 − 0.0459i)6-s − 0.0965·8-s + (0.915 − 0.401i)9-s + 0.0575·10-s + (0.241 − 0.417i)11-s + (−0.665 − 0.743i)12-s + (0.728 + 1.26i)13-s + (0.794 + 0.887i)15-s + (−0.496 + 0.859i)16-s + 1.58·17-s + (−0.00530 + 0.0480i)18-s − 0.436·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 - 0.690i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.722 - 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.722 - 0.690i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.722 - 0.690i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.989411 + 0.396734i\)
\(L(\frac12)\) \(\approx\) \(0.989411 + 0.396734i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.69 - 0.355i)T \)
7 \( 1 \)
good2 \( 1 + (0.0341 - 0.0592i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1.33 + 2.30i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.799 + 1.38i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.62 - 4.54i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 6.54T + 17T^{2} \)
19 \( 1 + 1.90T + 19T^{2} \)
23 \( 1 + (-1.53 - 2.65i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.19 - 5.53i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.35 - 5.81i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.22T + 37T^{2} \)
41 \( 1 + (-3.69 - 6.40i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.63 + 9.75i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.89 + 3.29i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 8.89T + 53T^{2} \)
59 \( 1 + (5.44 + 9.43i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.35 + 2.35i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.66 - 2.87i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.3T + 71T^{2} \)
73 \( 1 + 2.19T + 73T^{2} \)
79 \( 1 + (0.406 - 0.704i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.41 + 5.92i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 0.470T + 89T^{2} \)
97 \( 1 + (2.57 - 4.46i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.45869576602219914931811190550, −10.62020412506681314377597526585, −9.267784877619092839032112712145, −8.526316422105658669733035483825, −7.49851568309273526461632945022, −6.58546496794898999411560681852, −5.51162458822817179344839428777, −4.33921089610216573565566008641, −3.53951105441271418899147398241, −1.29642215745422083564642603501, 0.938690906841935525058547497117, 2.71876210308176637116123854823, 4.20750844259494960756133463147, 5.69162087093923836845220868638, 6.11932394831287316710645978566, 7.25045956166779584601078928341, 7.84445096675907817628657403576, 9.655298875336919269473995302573, 10.43823957560947268391747896032, 10.92902017785554314573553877723

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