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Properties

Label 2-2448-204.167-c0-0-1
Degree $2$
Conductor $2448$
Sign $0.935 + 0.354i$
Analytic cond. $1.22171$
Root an. cond. $1.10531$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.617 − 0.923i)5-s + (1.30 + 1.30i)13-s + i·17-s + (−0.0897 + 0.216i)25-s + (1.63 − 1.08i)29-s + (0.216 − 1.08i)37-s + (0.216 − 0.324i)41-s + (0.382 + 0.923i)49-s + (−0.707 − 1.70i)53-s + (1.08 − 1.63i)61-s + (0.400 − 2.01i)65-s + (−0.923 + 0.617i)73-s + (0.923 − 0.617i)85-s + (1.30 + 1.30i)89-s + (0.216 + 0.324i)97-s + ⋯
L(s)  = 1  + (−0.617 − 0.923i)5-s + (1.30 + 1.30i)13-s + i·17-s + (−0.0897 + 0.216i)25-s + (1.63 − 1.08i)29-s + (0.216 − 1.08i)37-s + (0.216 − 0.324i)41-s + (0.382 + 0.923i)49-s + (−0.707 − 1.70i)53-s + (1.08 − 1.63i)61-s + (0.400 − 2.01i)65-s + (−0.923 + 0.617i)73-s + (0.923 − 0.617i)85-s + (1.30 + 1.30i)89-s + (0.216 + 0.324i)97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2448\)    =    \(2^{4} \cdot 3^{2} \cdot 17\)
Sign: $0.935 + 0.354i$
Analytic conductor: \(1.22171\)
Root analytic conductor: \(1.10531\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2448} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2448,\ (\ :0),\ 0.935 + 0.354i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.147329711\)
\(L(\frac12)\) \(\approx\) \(1.147329711\)
\(L(1)\) 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 \( 1 - iT \)
good5 \( 1 + (0.617 + 0.923i)T + (-0.382 + 0.923i)T^{2} \)
7 \( 1 + (-0.382 - 0.923i)T^{2} \)
11 \( 1 + (0.923 - 0.382i)T^{2} \)
13 \( 1 + (-1.30 - 1.30i)T + iT^{2} \)
19 \( 1 + (-0.707 + 0.707i)T^{2} \)
23 \( 1 + (-0.923 + 0.382i)T^{2} \)
29 \( 1 + (-1.63 + 1.08i)T + (0.382 - 0.923i)T^{2} \)
31 \( 1 + (-0.923 - 0.382i)T^{2} \)
37 \( 1 + (-0.216 + 1.08i)T + (-0.923 - 0.382i)T^{2} \)
41 \( 1 + (-0.216 + 0.324i)T + (-0.382 - 0.923i)T^{2} \)
43 \( 1 + (-0.707 - 0.707i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
59 \( 1 + (-0.707 - 0.707i)T^{2} \)
61 \( 1 + (-1.08 + 1.63i)T + (-0.382 - 0.923i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (-0.923 - 0.382i)T^{2} \)
73 \( 1 + (0.923 - 0.617i)T + (0.382 - 0.923i)T^{2} \)
79 \( 1 + (-0.923 + 0.382i)T^{2} \)
83 \( 1 + (-0.707 + 0.707i)T^{2} \)
89 \( 1 + (-1.30 - 1.30i)T + iT^{2} \)
97 \( 1 + (-0.216 - 0.324i)T + (-0.382 + 0.923i)T^{2} \)
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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

−8.882235683542125042696705991920, −8.407095585098552595645616601828, −7.79106024448852834759949700464, −6.60878684725174419475855628784, −6.12821826629991583497891910313, −4.99823726058520233359585738729, −4.18180995001359548832104227616, −3.69957045319655624597739159724, −2.18131088645054117887161997970, −1.04231755241500774685889751831, 1.10858255417144799427874760840, 2.89281128111698171492605239121, 3.19145048059869305411042973961, 4.30270270903137345985307942762, 5.28757197978315521567492736682, 6.17555243473161388722997340562, 6.91472432795132160479330833630, 7.62037014051658221036249650533, 8.358127715857581837101304094430, 9.016380714181558434809147771790

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