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Properties

Label 2-21e2-49.15-c1-0-7
Degree $2$
Conductor $441$
Sign $-0.806 - 0.591i$
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  + (−0.603 + 2.64i)2-s + (−4.83 − 2.32i)4-s + (0.940 − 1.17i)5-s + (2.57 − 0.599i)7-s + (5.68 − 7.13i)8-s + (2.55 + 3.19i)10-s + (−0.818 + 3.58i)11-s + (−0.912 + 3.99i)13-s + (0.0297 + 7.17i)14-s + (8.74 + 10.9i)16-s + (−3.10 + 1.49i)17-s + 5.74·19-s + (−7.28 + 3.50i)20-s + (−8.99 − 4.33i)22-s + (3.61 + 1.74i)23-s + ⋯
L(s)  = 1  + (−0.426 + 1.87i)2-s + (−2.41 − 1.16i)4-s + (0.420 − 0.527i)5-s + (0.973 − 0.226i)7-s + (2.01 − 2.52i)8-s + (0.806 + 1.01i)10-s + (−0.246 + 1.08i)11-s + (−0.253 + 1.10i)13-s + (0.00794 + 1.91i)14-s + (2.18 + 2.74i)16-s + (−0.753 + 0.362i)17-s + 1.31·19-s + (−1.62 + 0.784i)20-s + (−1.91 − 0.923i)22-s + (0.753 + 0.362i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.806 - 0.591i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.806 - 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.806 - 0.591i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.806 - 0.591i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.341751 + 1.04373i\)
\(L(\frac12)\) \(\approx\) \(0.341751 + 1.04373i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-2.57 + 0.599i)T \)
good2 \( 1 + (0.603 - 2.64i)T + (-1.80 - 0.867i)T^{2} \)
5 \( 1 + (-0.940 + 1.17i)T + (-1.11 - 4.87i)T^{2} \)
11 \( 1 + (0.818 - 3.58i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (0.912 - 3.99i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 + (3.10 - 1.49i)T + (10.5 - 13.2i)T^{2} \)
19 \( 1 - 5.74T + 19T^{2} \)
23 \( 1 + (-3.61 - 1.74i)T + (14.3 + 17.9i)T^{2} \)
29 \( 1 + (-5.37 + 2.58i)T + (18.0 - 22.6i)T^{2} \)
31 \( 1 + 3.17T + 31T^{2} \)
37 \( 1 + (-0.851 + 0.410i)T + (23.0 - 28.9i)T^{2} \)
41 \( 1 + (-0.694 + 0.870i)T + (-9.12 - 39.9i)T^{2} \)
43 \( 1 + (-4.96 - 6.22i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (-0.237 + 1.04i)T + (-42.3 - 20.3i)T^{2} \)
53 \( 1 + (-2.81 - 1.35i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 + (4.89 + 6.13i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (3.90 - 1.87i)T + (38.0 - 47.6i)T^{2} \)
67 \( 1 + 6.17T + 67T^{2} \)
71 \( 1 + (3.76 + 1.81i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (-0.253 - 1.10i)T + (-65.7 + 31.6i)T^{2} \)
79 \( 1 + 15.6T + 79T^{2} \)
83 \( 1 + (2.63 + 11.5i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (-2.12 - 9.29i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 + 3.94T + 97T^{2} \)
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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

−11.36838409054381916957882552128, −10.07315587092547095344894632388, −9.295474916083458728278849558019, −8.696570997458574074297408417111, −7.53911961945376647568879780124, −7.13494417835095684777396980915, −5.91982474264901005105234170008, −4.89717786710981796242077292541, −4.45055737339450941257251216319, −1.48488582153952194105881722953, 0.951271690156370169624900465704, 2.50597422389011072535658043125, 3.17863356427787112544861898586, 4.66311925512416057277794507633, 5.59149196886534013974434876828, 7.49213734175566169323073976807, 8.471243562698634222255881294311, 9.108480467640126588962954544036, 10.28426834560889459200742150393, 10.73651960368146472192405985903

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