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Properties

Label 2-1205-5.4-c1-0-86
Degree $2$
Conductor $1205$
Sign $0.947 + 0.318i$
Analytic cond. $9.62197$
Root an. cond. $3.10193$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.49i·2-s − 1.66i·3-s − 4.22·4-s + (2.11 + 0.713i)5-s + 4.15·6-s − 4.24i·7-s − 5.56i·8-s + 0.229·9-s + (−1.78 + 5.28i)10-s − 6.39·11-s + 7.03i·12-s + 1.34i·13-s + 10.6·14-s + (1.18 − 3.52i)15-s + 5.42·16-s + 1.79i·17-s + ⋯
L(s)  = 1  + 1.76i·2-s − 0.960i·3-s − 2.11·4-s + (0.947 + 0.318i)5-s + 1.69·6-s − 1.60i·7-s − 1.96i·8-s + 0.0766·9-s + (−0.562 + 1.67i)10-s − 1.92·11-s + 2.03i·12-s + 0.371i·13-s + 2.83·14-s + (0.306 − 0.910i)15-s + 1.35·16-s + 0.436i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $0.947 + 0.318i$
Analytic conductor: \(9.62197\)
Root analytic conductor: \(3.10193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1205} (724, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1205,\ (\ :1/2),\ 0.947 + 0.318i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.231912585\)
\(L(\frac12)\) \(\approx\) \(1.231912585\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.11 - 0.713i)T \)
241 \( 1 - T \)
good2 \( 1 - 2.49iT - 2T^{2} \)
3 \( 1 + 1.66iT - 3T^{2} \)
7 \( 1 + 4.24iT - 7T^{2} \)
11 \( 1 + 6.39T + 11T^{2} \)
13 \( 1 - 1.34iT - 13T^{2} \)
17 \( 1 - 1.79iT - 17T^{2} \)
19 \( 1 - 3.80T + 19T^{2} \)
23 \( 1 + 5.33iT - 23T^{2} \)
29 \( 1 + 1.62T + 29T^{2} \)
31 \( 1 + 3.96T + 31T^{2} \)
37 \( 1 + 10.3iT - 37T^{2} \)
41 \( 1 - 2.25T + 41T^{2} \)
43 \( 1 + 1.94iT - 43T^{2} \)
47 \( 1 + 11.9iT - 47T^{2} \)
53 \( 1 + 10.9iT - 53T^{2} \)
59 \( 1 + 9.06T + 59T^{2} \)
61 \( 1 - 14.1T + 61T^{2} \)
67 \( 1 + 4.44iT - 67T^{2} \)
71 \( 1 + 6.47T + 71T^{2} \)
73 \( 1 - 4.44iT - 73T^{2} \)
79 \( 1 - 9.81T + 79T^{2} \)
83 \( 1 - 14.6iT - 83T^{2} \)
89 \( 1 + 10.4T + 89T^{2} \)
97 \( 1 - 5.74iT - 97T^{2} \)
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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

−9.596122504113487707822065727659, −8.389972290646521264173056032593, −7.69525269842464895406951120886, −7.12089026604088273317211938590, −6.72686959813392096042058378817, −5.69080950557535166017059068193, −5.02655773264439605435817943753, −3.87087890687793302766815818448, −2.17210392769934774423813320699, −0.52745356082357524971387140857, 1.56582295555071590509591516011, 2.71684918335530251061743476076, 3.08459449956833720305817485474, 4.64855500116390292374632873489, 5.24297111771157448650296238684, 5.73898696652167977098847702189, 7.72810057764406647989966916026, 8.764022778402806504681556400693, 9.451416107846933364171858023191, 9.786008391241019418436018153887

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