Properties

Label 2-303450-1.1-c1-0-17
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 $0$

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 + 6·11-s − 12-s − 4·13-s + 14-s + 16-s − 18-s − 19-s + 21-s − 6·22-s + 24-s + 4·26-s − 27-s − 28-s − 9·29-s + 4·31-s − 32-s − 6·33-s + 36-s − 2·37-s + 38-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 + 1.80·11-s − 0.288·12-s − 1.10·13-s + 0.267·14-s + 1/4·16-s − 0.235·18-s − 0.229·19-s + 0.218·21-s − 1.27·22-s + 0.204·24-s + 0.784·26-s − 0.192·27-s − 0.188·28-s − 1.67·29-s + 0.718·31-s − 0.176·32-s − 1.04·33-s + 1/6·36-s − 0.328·37-s + 0.162·38-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: $\chi_{303450} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 303450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8923737871\)
\(L(\frac12)\) \(\approx\) \(0.8923737871\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−12.34522984949742, −12.21988206776388, −11.73175010444377, −11.31849707785564, −10.89755951368868, −10.29179543889712, −9.842305206377697, −9.524386466035799, −9.098485936164726, −8.656715938526615, −8.050871450529008, −7.468381411321856, −7.005480654265039, −6.696184611314908, −6.251324848481814, −5.678007541049300, −5.195195488808972, −4.532832877995930, −4.005980414178942, −3.529101633938977, −2.905846850863633, −2.084894326688956, −1.714342792312488, −1.003275358683091, −0.3322443537091680, 0.3322443537091680, 1.003275358683091, 1.714342792312488, 2.084894326688956, 2.905846850863633, 3.529101633938977, 4.005980414178942, 4.532832877995930, 5.195195488808972, 5.678007541049300, 6.251324848481814, 6.696184611314908, 7.005480654265039, 7.468381411321856, 8.050871450529008, 8.656715938526615, 9.098485936164726, 9.524386466035799, 9.842305206377697, 10.29179543889712, 10.89755951368868, 11.31849707785564, 11.73175010444377, 12.21988206776388, 12.34522984949742

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