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Properties

Label 2-2352-7.4-c1-0-9
Degree $2$
Conductor $2352$
Sign $-0.605 - 0.795i$
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 + (−2 + 3.46i)11-s + 6·13-s − 1.99·15-s + (−1 + 1.73i)17-s + (−2 − 3.46i)19-s + (4 + 6.92i)23-s + (0.500 − 0.866i)25-s + 0.999·27-s − 2·29-s + (−1.99 − 3.46i)33-s + (5 + 8.66i)37-s + (−3 + 5.19i)39-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (0.447 + 0.774i)5-s + (−0.166 − 0.288i)9-s + (−0.603 + 1.04i)11-s + 1.66·13-s − 0.516·15-s + (−0.242 + 0.420i)17-s + (−0.458 − 0.794i)19-s + (0.834 + 1.44i)23-s + (0.100 − 0.173i)25-s + 0.192·27-s − 0.371·29-s + (−0.348 − 0.603i)33-s + (0.821 + 1.42i)37-s + (−0.480 + 0.832i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $-0.605 - 0.795i$
Analytic conductor: \(18.7808\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (1537, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.548570267\)
\(L(\frac12)\) \(\approx\) \(1.548570267\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 6T + 13T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4 - 6.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5 - 8.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \)
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 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (5 - 8.66i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−9.368475444754522374718561456056, −8.590540788984670057295091452881, −7.67891367640837341167791734360, −6.74825564270252250658741036469, −6.23089322094063686615257245394, −5.32903068299519592921866715257, −4.48827772370948461906822379338, −3.53308399213546321049714109111, −2.64478692983707578553363107074, −1.44392184869960712883829266041, 0.57515332802897908637646260890, 1.54300822451132350383270739731, 2.75673251700723453675903034931, 3.84002387962593328066017770461, 4.87543922481801154558251501036, 5.78831819601328764971785715378, 6.12167169919231988386940114456, 7.12732579585037915469358335368, 8.153532025306567319842610407143, 8.660913775422952447732167881809

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