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Properties

Label 2-1224-17.15-c1-0-13
Degree $2$
Conductor $1224$
Sign $0.168 + 0.985i$
Analytic cond. $9.77368$
Root an. cond. $3.12629$
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  + (−1.70 + 0.707i)5-s + (−0.292 − 0.121i)7-s + (−0.292 + 0.707i)11-s + (−1 − 4i)17-s + (−1.58 − 1.58i)19-s + (2.53 − 6.12i)23-s + (−1.12 + 1.12i)25-s + (5.12 − 2.12i)29-s + (0.878 + 2.12i)31-s + 0.585·35-s + (0.292 + 0.707i)37-s + (2.29 + 0.949i)41-s + (7.24 − 7.24i)43-s − 12.8i·47-s + (−4.87 − 4.87i)49-s + ⋯
L(s)  = 1  + (−0.763 + 0.316i)5-s + (−0.110 − 0.0458i)7-s + (−0.0883 + 0.213i)11-s + (−0.242 − 0.970i)17-s + (−0.363 − 0.363i)19-s + (0.528 − 1.27i)23-s + (−0.224 + 0.224i)25-s + (0.951 − 0.393i)29-s + (0.157 + 0.381i)31-s + 0.0990·35-s + (0.0481 + 0.116i)37-s + (0.358 + 0.148i)41-s + (1.10 − 1.10i)43-s − 1.87i·47-s + (−0.696 − 0.696i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.168 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.168 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $0.168 + 0.985i$
Analytic conductor: \(9.77368\)
Root analytic conductor: \(3.12629\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1224,\ (\ :1/2),\ 0.168 + 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9797605671\)
\(L(\frac12)\) \(\approx\) \(0.9797605671\)
\(L(\frac{3}{2})\) 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 \( 1 \)
17 \( 1 + (1 + 4i)T \)
good5 \( 1 + (1.70 - 0.707i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (0.292 + 0.121i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (0.292 - 0.707i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 - 13T^{2} \)
19 \( 1 + (1.58 + 1.58i)T + 19iT^{2} \)
23 \( 1 + (-2.53 + 6.12i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-5.12 + 2.12i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (-0.878 - 2.12i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (-0.292 - 0.707i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-2.29 - 0.949i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (-7.24 + 7.24i)T - 43iT^{2} \)
47 \( 1 + 12.8iT - 47T^{2} \)
53 \( 1 + (2.17 + 2.17i)T + 53iT^{2} \)
59 \( 1 + (5.58 - 5.58i)T - 59iT^{2} \)
61 \( 1 + (9.12 + 3.77i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 - 9.65T + 67T^{2} \)
71 \( 1 + (3.70 + 8.94i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (-7.36 + 3.05i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (-5.46 + 13.1i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (-7.24 - 7.24i)T + 83iT^{2} \)
89 \( 1 + 1.65iT - 89T^{2} \)
97 \( 1 + (14.7 - 6.12i)T + (68.5 - 68.5i)T^{2} \)
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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

−9.499037831137933121799181237423, −8.698617590768325606977836205455, −7.88164658837448436989452813816, −7.03981082642834825427613619104, −6.44044221639655996600013158558, −5.12994757182802879160494629000, −4.37060383723631833133911777630, −3.31266452640627069092739155522, −2.31783353653941187022356387137, −0.45357221592682552465432935919, 1.28284709519217437641450148543, 2.81048418647478911601288565581, 3.88974877315146209918453287680, 4.61868735727299106387286265243, 5.78070122047410948872158415983, 6.52723517245253920413130694058, 7.72495111185269274742728295730, 8.105836832441310819189458381081, 9.082455125109831132436982473981, 9.804829853171364190525951714096

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