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Properties

Label 2-1452-11.3-c1-0-6
Degree $2$
Conductor $1452$
Sign $0.859 - 0.511i$
Analytic cond. $11.5942$
Root an. cond. $3.40503$
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  + (0.309 + 0.951i)3-s + (−2.42 − 1.76i)5-s + (−0.618 + 1.90i)7-s + (−0.809 + 0.587i)9-s + (4.04 − 2.93i)13-s + (0.927 − 2.85i)15-s + (−2.42 − 1.76i)17-s + (1.23 + 3.80i)19-s − 1.99·21-s + 6·23-s + (1.23 + 3.80i)25-s + (−0.809 − 0.587i)27-s + (2.78 − 8.55i)29-s + (−6.47 + 4.70i)31-s + (4.85 − 3.52i)35-s + ⋯
L(s)  = 1  + (0.178 + 0.549i)3-s + (−1.08 − 0.788i)5-s + (−0.233 + 0.718i)7-s + (−0.269 + 0.195i)9-s + (1.12 − 0.815i)13-s + (0.239 − 0.736i)15-s + (−0.588 − 0.427i)17-s + (0.283 + 0.872i)19-s − 0.436·21-s + 1.25·23-s + (0.247 + 0.760i)25-s + (−0.155 − 0.113i)27-s + (0.516 − 1.58i)29-s + (−1.16 + 0.844i)31-s + (0.820 − 0.596i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1452\)    =    \(2^{2} \cdot 3 \cdot 11^{2}\)
Sign: $0.859 - 0.511i$
Analytic conductor: \(11.5942\)
Root analytic conductor: \(3.40503\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1452} (1213, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1452,\ (\ :1/2),\ 0.859 - 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.382336511\)
\(L(\frac12)\) \(\approx\) \(1.382336511\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 \)
good5 \( 1 + (2.42 + 1.76i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.618 - 1.90i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-4.04 + 2.93i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.42 + 1.76i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.23 - 3.80i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 + (-2.78 + 8.55i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (6.47 - 4.70i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.16 - 6.65i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.927 - 2.85i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 10T + 43T^{2} \)
47 \( 1 + (-3.70 - 11.4i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-7.28 + 5.29i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.85 - 5.70i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-11.3 - 8.22i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 + (-4.85 - 3.52i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-3.09 + 9.51i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-6.47 + 4.70i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 3T + 89T^{2} \)
97 \( 1 + (-0.809 + 0.587i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.357205213118542504105166600308, −8.751911374919635387495354960786, −8.206259915157251589407213361142, −7.39781651019100193865909353382, −6.12956358569672344448094232947, −5.35949056513269011364636127314, −4.43827703278067998480858643630, −3.65110855983992281587761075043, −2.71416032933528340367741637675, −0.925138583538617499636401331588, 0.77247864324304186452758042559, 2.31368331292170652345724285183, 3.60854967774636563514110593867, 3.92865717907406765128193357065, 5.32324097444527499570591065336, 6.64859182150183993533122577816, 7.00121142370598595417228173046, 7.59643924515528238659884926248, 8.728961724604837475648112079803, 9.129523265142507617641328049412

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