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Properties

Label 2-2352-7.2-c1-0-13
Degree $2$
Conductor $2352$
Sign $-0.386 - 0.922i$
Analytic cond. $18.7808$
Root an. cond. $4.33368$
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 + (−0.499 + 0.866i)9-s + (1 + 1.73i)11-s + 3·13-s − 1.99·15-s + (4 + 6.92i)17-s + (0.5 − 0.866i)19-s + (4 − 6.92i)23-s + (0.500 + 0.866i)25-s − 0.999·27-s + 4·29-s + (−1.5 − 2.59i)31-s + (−0.999 + 1.73i)33-s + (0.5 − 0.866i)37-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.447 + 0.774i)5-s + (−0.166 + 0.288i)9-s + (0.301 + 0.522i)11-s + 0.832·13-s − 0.516·15-s + (0.970 + 1.68i)17-s + (0.114 − 0.198i)19-s + (0.834 − 1.44i)23-s + (0.100 + 0.173i)25-s − 0.192·27-s + 0.742·29-s + (−0.269 − 0.466i)31-s + (−0.174 + 0.301i)33-s + (0.0821 − 0.142i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $-0.386 - 0.922i$
Analytic conductor: \(18.7808\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ -0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.840793851\)
\(L(\frac12)\) \(\approx\) \(1.840793851\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 3T + 13T^{2} \)
17 \( 1 + (-4 - 6.92i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4 + 6.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + (1.5 + 2.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 11T + 43T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6 - 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3 - 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.5 - 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10T + 71T^{2} \)
73 \( 1 + (5.5 + 9.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.5 - 2.59i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 2T + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−9.143264662367284156491733831686, −8.435835724499907833144036077395, −7.83075999805203006324136357550, −6.82867239440940188102734645882, −6.27419268722651904757130636951, −5.20956460509963210842480854915, −4.20824694756349333236988631482, −3.55176317248916247719592551282, −2.73907338823521987827081222183, −1.39303524145346389800378997993, 0.67428079401453006570184899168, 1.54792793667599482127469856975, 3.10376436160811574197808181928, 3.60835224138370136075431753142, 4.93937392355020217574295133954, 5.43065355562205149860590767182, 6.59642910389628417750054398055, 7.21275613423167263104332677934, 8.177144798429578610070363783151, 8.539412989134847386927295035394

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