Properties

Degree $2$
Conductor $7350$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s + 2·11-s − 12-s + 2·13-s + 16-s − 4·17-s − 18-s − 2·22-s − 8·23-s + 24-s − 2·26-s − 27-s + 2·31-s − 32-s − 2·33-s + 4·34-s + 36-s − 8·37-s − 2·39-s + 2·41-s + 2·43-s + 2·44-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.603·11-s − 0.288·12-s + 0.554·13-s + 1/4·16-s − 0.970·17-s − 0.235·18-s − 0.426·22-s − 1.66·23-s + 0.204·24-s − 0.392·26-s − 0.192·27-s + 0.359·31-s − 0.176·32-s − 0.348·33-s + 0.685·34-s + 1/6·36-s − 1.31·37-s − 0.320·39-s + 0.312·41-s + 0.304·43-s + 0.301·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7350\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{7350} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7350,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9584112699\)
\(L(\frac12)\) \(\approx\) \(0.9584112699\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−17.30105323732160, −16.49664219005953, −16.09885908680257, −15.52765153132769, −15.01136846668193, −14.01855743751627, −13.71984066596780, −12.81903213522003, −12.06171734589444, −11.81827284266811, −10.94364768441175, −10.59755286838846, −9.876569812000490, −9.195113903251254, −8.633371845034471, −7.952472149658223, −7.184976165340548, −6.499959306080130, −6.047326315488616, −5.236396493588644, −4.226097559653089, −3.679867946375541, −2.421333898805811, −1.647389087865914, −0.5730785581451872, 0.5730785581451872, 1.647389087865914, 2.421333898805811, 3.679867946375541, 4.226097559653089, 5.236396493588644, 6.047326315488616, 6.499959306080130, 7.184976165340548, 7.952472149658223, 8.633371845034471, 9.195113903251254, 9.876569812000490, 10.59755286838846, 10.94364768441175, 11.81827284266811, 12.06171734589444, 12.81903213522003, 13.71984066596780, 14.01855743751627, 15.01136846668193, 15.52765153132769, 16.09885908680257, 16.49664219005953, 17.30105323732160

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