Properties

Label 2-10304-1.1-c1-0-1
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 − 6·17-s + 6·19-s + 2·21-s − 23-s − 5·25-s + 4·27-s + 6·29-s − 8·31-s − 2·37-s − 2·41-s − 8·43-s + 8·47-s + 49-s + 12·51-s + 2·53-s − 12·57-s − 6·59-s − 63-s − 12·67-s + 2·69-s + 8·71-s − 6·73-s + 10·75-s + 16·79-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.377·7-s + 1/3·9-s − 1.45·17-s + 1.37·19-s + 0.436·21-s − 0.208·23-s − 25-s + 0.769·27-s + 1.11·29-s − 1.43·31-s − 0.328·37-s − 0.312·41-s − 1.21·43-s + 1.16·47-s + 1/7·49-s + 1.68·51-s + 0.274·53-s − 1.58·57-s − 0.781·59-s − 0.125·63-s − 1.46·67-s + 0.240·69-s + 0.949·71-s − 0.702·73-s + 1.15·75-s + 1.80·79-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

\(L(1)\) \(\approx\) \(0.6381734044\)
\(L(\frac12)\) \(\approx\) \(0.6381734044\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
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

−16.46753370536440, −16.21714782236016, −15.55401276900372, −15.11924706818992, −14.15987317733247, −13.65109017999799, −13.18379534965632, −12.32770426383971, −11.90427594103538, −11.49169691514433, −10.69410055831462, −10.43427769286864, −9.486488263686222, −9.082098351540372, −8.235610494898363, −7.465957842846244, −6.759075048254671, −6.315964658007927, −5.545757619190799, −5.092190274657899, −4.282952096898660, −3.477681673587700, −2.571535099520529, −1.547729596637332, −0.4021030751198909, 0.4021030751198909, 1.547729596637332, 2.571535099520529, 3.477681673587700, 4.282952096898660, 5.092190274657899, 5.545757619190799, 6.315964658007927, 6.759075048254671, 7.465957842846244, 8.235610494898363, 9.082098351540372, 9.486488263686222, 10.43427769286864, 10.69410055831462, 11.49169691514433, 11.90427594103538, 12.32770426383971, 13.18379534965632, 13.65109017999799, 14.15987317733247, 15.11924706818992, 15.55401276900372, 16.21714782236016, 16.46753370536440

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