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Properties

Label 2-2303-2303.1409-c0-0-1
Degree $2$
Conductor $2303$
Sign $0.218 - 0.975i$
Analytic cond. $1.14934$
Root an. cond. $1.07207$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.172 + 0.117i)2-s + (−1.37 + 1.27i)3-s + (−0.349 + 0.890i)4-s + (0.0871 − 0.381i)6-s + (−0.337 − 0.941i)7-s + (−0.0910 − 0.398i)8-s + (0.187 − 2.49i)9-s + (−0.654 − 1.66i)12-s + (0.169 + 0.122i)14-s + (−0.638 − 0.592i)16-s + (1.72 + 0.260i)17-s + (0.261 + 0.453i)18-s + (1.66 + 0.862i)21-s + (0.632 + 0.431i)24-s + (0.826 + 0.563i)25-s + ⋯
L(s)  = 1  + (−0.172 + 0.117i)2-s + (−1.37 + 1.27i)3-s + (−0.349 + 0.890i)4-s + (0.0871 − 0.381i)6-s + (−0.337 − 0.941i)7-s + (−0.0910 − 0.398i)8-s + (0.187 − 2.49i)9-s + (−0.654 − 1.66i)12-s + (0.169 + 0.122i)14-s + (−0.638 − 0.592i)16-s + (1.72 + 0.260i)17-s + (0.261 + 0.453i)18-s + (1.66 + 0.862i)21-s + (0.632 + 0.431i)24-s + (0.826 + 0.563i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.218 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.218 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2303\)    =    \(7^{2} \cdot 47\)
Sign: $0.218 - 0.975i$
Analytic conductor: \(1.14934\)
Root analytic conductor: \(1.07207\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2303} (1409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2303,\ (\ :0),\ 0.218 - 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5716206120\)
\(L(\frac12)\) \(\approx\) \(0.5716206120\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.337 + 0.941i)T \)
47 \( 1 + (-0.826 + 0.563i)T \)
good2 \( 1 + (0.172 - 0.117i)T + (0.365 - 0.930i)T^{2} \)
3 \( 1 + (1.37 - 1.27i)T + (0.0747 - 0.997i)T^{2} \)
5 \( 1 + (-0.826 - 0.563i)T^{2} \)
11 \( 1 + (0.988 - 0.149i)T^{2} \)
13 \( 1 + (-0.623 - 0.781i)T^{2} \)
17 \( 1 + (-1.72 - 0.260i)T + (0.955 + 0.294i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.955 + 0.294i)T^{2} \)
29 \( 1 + (0.222 + 0.974i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.204 + 0.522i)T + (-0.733 + 0.680i)T^{2} \)
41 \( 1 + (0.900 - 0.433i)T^{2} \)
43 \( 1 + (0.900 + 0.433i)T^{2} \)
53 \( 1 + (-0.648 + 1.65i)T + (-0.733 - 0.680i)T^{2} \)
59 \( 1 + (1.80 - 0.557i)T + (0.826 - 0.563i)T^{2} \)
61 \( 1 + (-0.384 - 0.978i)T + (-0.733 + 0.680i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.204 + 0.256i)T + (-0.222 + 0.974i)T^{2} \)
73 \( 1 + (-0.365 - 0.930i)T^{2} \)
79 \( 1 + (-0.995 + 1.72i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.658 - 0.317i)T + (0.623 - 0.781i)T^{2} \)
89 \( 1 + (0.120 - 1.61i)T + (-0.988 - 0.149i)T^{2} \)
97 \( 1 - 1.82T + T^{2} \)
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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

−9.488472250931856005008853474313, −8.823761437740689041602989653609, −7.69199005266814866807652722066, −7.03780081035476228952489202886, −6.15150154106095815609639830928, −5.28785988537439604073227531730, −4.56705168577148153331148561182, −3.70906905872057979738853486862, −3.30586821234855672110767895771, −0.808852623893024233265934016311, 0.820020427480292288520595349255, 1.76071345937832225137669846779, 2.87035488338021156919512202767, 4.67843853474262938187864584983, 5.37429864088681339134970920598, 5.91955222882257028349749557057, 6.44457217708239368928746370609, 7.36388307423098594728527641505, 8.155978157391347846252688679690, 9.051916737559388340170560036896

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