Properties

Label 2-10304-1.1-c1-0-5
Degree $2$
Conductor $10304$
Sign $1$
Analytic cond. $82.2778$
Root an. cond. $9.07071$
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  − 2·3-s + 7-s + 9-s − 4·11-s + 6·17-s + 6·19-s − 2·21-s − 23-s − 5·25-s + 4·27-s − 10·29-s + 4·31-s + 8·33-s + 2·37-s − 10·41-s + 4·43-s + 12·47-s + 49-s − 12·51-s + 6·53-s − 12·57-s + 2·59-s + 63-s + 2·69-s − 8·71-s − 6·73-s + 10·75-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.377·7-s + 1/3·9-s − 1.20·11-s + 1.45·17-s + 1.37·19-s − 0.436·21-s − 0.208·23-s − 25-s + 0.769·27-s − 1.85·29-s + 0.718·31-s + 1.39·33-s + 0.328·37-s − 1.56·41-s + 0.609·43-s + 1.75·47-s + 1/7·49-s − 1.68·51-s + 0.824·53-s − 1.58·57-s + 0.260·59-s + 0.125·63-s + 0.240·69-s − 0.949·71-s − 0.702·73-s + 1.15·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10304\)    =    \(2^{6} \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(82.2778\)
Root analytic conductor: \(9.07071\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{10304} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 10304,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.032532408\)
\(L(\frac12)\) \(\approx\) \(1.032532408\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - T \)
23 \( 1 + T \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−16.66653833748048, −16.13176526292533, −15.57804439035923, −15.00390258043957, −14.26044795110267, −13.65644623940532, −13.16801175841340, −12.28208011336519, −11.94557715734098, −11.44862740646688, −10.82084710589718, −10.18015155885083, −9.818420400571708, −8.929744857731069, −8.035232285104246, −7.590424338807004, −7.060184480036594, −5.922228652347161, −5.525788772251531, −5.279781597878424, −4.304385786051079, −3.419995600869724, −2.597806433376867, −1.490895407789185, −0.5288837173404470, 0.5288837173404470, 1.490895407789185, 2.597806433376867, 3.419995600869724, 4.304385786051079, 5.279781597878424, 5.525788772251531, 5.922228652347161, 7.060184480036594, 7.590424338807004, 8.035232285104246, 8.929744857731069, 9.818420400571708, 10.18015155885083, 10.82084710589718, 11.44862740646688, 11.94557715734098, 12.28208011336519, 13.16801175841340, 13.65644623940532, 14.26044795110267, 15.00390258043957, 15.57804439035923, 16.13176526292533, 16.66653833748048

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