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Properties

Label 2-1248-104.99-c1-0-4
Degree $2$
Conductor $1248$
Sign $-0.423 - 0.906i$
Analytic cond. $9.96533$
Root an. cond. $3.15679$
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  + 3-s + (−1.67 + 1.67i)5-s + (1.16 + 1.16i)7-s + 9-s + (0.391 − 0.391i)11-s + (−3.59 + 0.311i)13-s + (−1.67 + 1.67i)15-s + 2.95i·17-s + (0.785 + 0.785i)19-s + (1.16 + 1.16i)21-s − 2.14·23-s − 0.591i·25-s + 27-s + 9.69i·29-s + (−1.16 + 1.16i)31-s + ⋯
L(s)  = 1  + 0.577·3-s + (−0.747 + 0.747i)5-s + (0.438 + 0.438i)7-s + 0.333·9-s + (0.118 − 0.118i)11-s + (−0.996 + 0.0863i)13-s + (−0.431 + 0.431i)15-s + 0.717i·17-s + (0.180 + 0.180i)19-s + (0.253 + 0.253i)21-s − 0.447·23-s − 0.118i·25-s + 0.192·27-s + 1.80i·29-s + (−0.208 + 0.208i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.423 - 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.423 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1248\)    =    \(2^{5} \cdot 3 \cdot 13\)
Sign: $-0.423 - 0.906i$
Analytic conductor: \(9.96533\)
Root analytic conductor: \(3.15679\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1248} (463, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1248,\ (\ :1/2),\ -0.423 - 0.906i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.348435620\)
\(L(\frac12)\) \(\approx\) \(1.348435620\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
13 \( 1 + (3.59 - 0.311i)T \)
good5 \( 1 + (1.67 - 1.67i)T - 5iT^{2} \)
7 \( 1 + (-1.16 - 1.16i)T + 7iT^{2} \)
11 \( 1 + (-0.391 + 0.391i)T - 11iT^{2} \)
17 \( 1 - 2.95iT - 17T^{2} \)
19 \( 1 + (-0.785 - 0.785i)T + 19iT^{2} \)
23 \( 1 + 2.14T + 23T^{2} \)
29 \( 1 - 9.69iT - 29T^{2} \)
31 \( 1 + (1.16 - 1.16i)T - 31iT^{2} \)
37 \( 1 + (-1.87 - 1.87i)T + 37iT^{2} \)
41 \( 1 + (1.21 + 1.21i)T + 41iT^{2} \)
43 \( 1 + 6.64iT - 43T^{2} \)
47 \( 1 + (5.19 + 5.19i)T + 47iT^{2} \)
53 \( 1 - 13.9iT - 53T^{2} \)
59 \( 1 + (8.86 - 8.86i)T - 59iT^{2} \)
61 \( 1 + 5.77iT - 61T^{2} \)
67 \( 1 + (-5.85 - 5.85i)T + 67iT^{2} \)
71 \( 1 + (5.33 - 5.33i)T - 71iT^{2} \)
73 \( 1 + (5.90 - 5.90i)T - 73iT^{2} \)
79 \( 1 + 1.82iT - 79T^{2} \)
83 \( 1 + (3.29 + 3.29i)T + 83iT^{2} \)
89 \( 1 + (7.48 - 7.48i)T - 89iT^{2} \)
97 \( 1 + (-4.94 - 4.94i)T + 97iT^{2} \)
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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

−10.02824991539856300973978143558, −8.966620393133058413357163391056, −8.371170024727315267200024272734, −7.43751990169258419254294070116, −6.99500423409337363667171446314, −5.76856816598066565262535409445, −4.74157627950154060781260941302, −3.71050564515569268953201193121, −2.90521545773001216137055599457, −1.75081643614830688675496902219, 0.51893068629086243624367304446, 2.04279063928939828352881094759, 3.25787184551411615916917209015, 4.47146511904019415742072068350, 4.73644930098844318112921479683, 6.13377951676081460297548966599, 7.35589878149653385880662555651, 7.81124010040046773151725225079, 8.474943570699964814072480640933, 9.529931498247610370114433691492

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