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Properties

Label 2-303450-1.1-c1-0-100
Degree $2$
Conductor $303450$
Sign $-1$
Analytic cond. $2423.06$
Root an. cond. $49.2245$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s + 12-s − 2·13-s − 14-s + 16-s − 18-s − 4·19-s + 21-s − 24-s + 2·26-s + 27-s + 28-s − 6·29-s − 8·31-s − 32-s + 36-s + 2·37-s + 4·38-s − 2·39-s − 6·41-s − 42-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.288·12-s − 0.554·13-s − 0.267·14-s + 1/4·16-s − 0.235·18-s − 0.917·19-s + 0.218·21-s − 0.204·24-s + 0.392·26-s + 0.192·27-s + 0.188·28-s − 1.11·29-s − 1.43·31-s − 0.176·32-s + 1/6·36-s + 0.328·37-s + 0.648·38-s − 0.320·39-s − 0.937·41-s − 0.154·42-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(303450\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(2423.06\)
Root analytic conductor: \(49.2245\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 303450,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
17 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−12.85409146478262, −12.62222128194532, −11.80724066831354, −11.50075013891550, −11.00425312104086, −10.59452751440588, −10.08463007932870, −9.539523959407926, −9.340824038535924, −8.643310064861815, −8.415248593108721, −7.806409803609159, −7.501609581356817, −6.929041900546463, −6.558400820688335, −5.885139648039986, −5.347815917294974, −4.837202740404518, −4.230563364331510, −3.633556455625106, −3.206421779714820, −2.459736727064929, −1.902347795761095, −1.689967696106943, −0.6944216974849955, 0, 0.6944216974849955, 1.689967696106943, 1.902347795761095, 2.459736727064929, 3.206421779714820, 3.633556455625106, 4.230563364331510, 4.837202740404518, 5.347815917294974, 5.885139648039986, 6.558400820688335, 6.929041900546463, 7.501609581356817, 7.806409803609159, 8.415248593108721, 8.643310064861815, 9.340824038535924, 9.539523959407926, 10.08463007932870, 10.59452751440588, 11.00425312104086, 11.50075013891550, 11.80724066831354, 12.62222128194532, 12.85409146478262

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