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 + 6·13-s + 16-s + 2·17-s + 18-s + 2·22-s + 4·23-s − 24-s + 6·26-s − 27-s + 8·31-s + 32-s − 2·33-s + 2·34-s + 36-s − 2·37-s − 6·39-s − 2·41-s + 4·43-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 + 1.66·13-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.426·22-s + 0.834·23-s − 0.204·24-s + 1.17·26-s − 0.192·27-s + 1.43·31-s + 0.176·32-s − 0.348·33-s + 0.342·34-s + 1/6·36-s − 0.328·37-s − 0.960·39-s − 0.312·41-s + 0.609·43-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: $\chi_{7350} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7350,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.364639067\)
\(L(\frac12)\) \(\approx\) \(3.364639067\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−7.87531176933172690232251758798, −6.86015454229513548668779514708, −6.42628017020170686836520225338, −5.86080294740503652248590880696, −5.08287292126618395699564211307, −4.38228326803548254663203186818, −3.61850544170667382974204431350, −2.97800548216663498952436660758, −1.66206688865020159921577692551, −0.934321117792836057923013236710, 0.934321117792836057923013236710, 1.66206688865020159921577692551, 2.97800548216663498952436660758, 3.61850544170667382974204431350, 4.38228326803548254663203186818, 5.08287292126618395699564211307, 5.86080294740503652248590880696, 6.42628017020170686836520225338, 6.86015454229513548668779514708, 7.87531176933172690232251758798

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