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Properties

Label 2-744-31.25-c1-0-2
Degree $2$
Conductor $744$
Sign $-0.695 - 0.718i$
Analytic cond. $5.94086$
Root an. cond. $2.43738$
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  + (0.5 + 0.866i)3-s + (−1 + 1.73i)5-s + (2.5 + 4.33i)7-s + (−0.499 + 0.866i)9-s + (−1.5 + 2.59i)11-s − 1.99·15-s + (−3 − 5.19i)17-s + (−3 − 5.19i)19-s + (−2.5 + 4.33i)21-s + (0.500 + 0.866i)25-s − 0.999·27-s − 3·29-s + (3.5 − 4.33i)31-s − 3·33-s − 10·35-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.447 + 0.774i)5-s + (0.944 + 1.63i)7-s + (−0.166 + 0.288i)9-s + (−0.452 + 0.783i)11-s − 0.516·15-s + (−0.727 − 1.26i)17-s + (−0.688 − 1.19i)19-s + (−0.545 + 0.944i)21-s + (0.100 + 0.173i)25-s − 0.192·27-s − 0.557·29-s + (0.628 − 0.777i)31-s − 0.522·33-s − 1.69·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.695 - 0.718i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(744\)    =    \(2^{3} \cdot 3 \cdot 31\)
Sign: $-0.695 - 0.718i$
Analytic conductor: \(5.94086\)
Root analytic conductor: \(2.43738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{744} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 744,\ (\ :1/2),\ -0.695 - 0.718i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.549262 + 1.29506i\)
\(L(\frac12)\) \(\approx\) \(0.549262 + 1.29506i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (-3.5 + 4.33i)T \)
good5 \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2.5 - 4.33i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3 + 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
37 \( 1 + (-4 - 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2 - 3.46i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4 - 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 + (0.5 - 0.866i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.5 - 9.52i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 + (-6 + 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-8 + 13.8i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.5 - 9.52i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 - 19T + 97T^{2} \)
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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

−10.94224631653692284417670356034, −9.634346238744087288972001398414, −9.060456515540409102174226167427, −8.156754132713043986315727172760, −7.35655548288506403881921642118, −6.27679101467308189855245067470, −5.04620707468381872846199176167, −4.51714124980889238160923728524, −2.83359504687250868058375465329, −2.31696326690223173321953950990, 0.69725168824908660336167606506, 1.89708252630956332401921341291, 3.77933536172909695221875298239, 4.29232204018706722929680766732, 5.53937765523541679275364713373, 6.69170328316762355382721325835, 7.69790526113818724892872669746, 8.224831336369861050783376517836, 8.818312941370086560876182168054, 10.37531712484680014223821390151

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