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Properties

Label 2-1265-1265.1022-c0-0-1
Degree $2$
Conductor $1265$
Sign $0.912 + 0.408i$
Analytic cond. $0.631317$
Root an. cond. $0.794554$
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  + (−0.936 + 0.203i)3-s + (−0.989 − 0.142i)4-s + (−0.909 + 0.415i)5-s + (−0.0739 + 0.0337i)9-s + (−0.755 − 0.654i)11-s + (0.956 − 0.0683i)12-s + (0.767 − 0.574i)15-s + (0.959 + 0.281i)16-s + (0.959 − 0.281i)20-s + (0.755 + 0.654i)23-s + (0.654 − 0.755i)25-s + (0.829 − 0.621i)27-s + (1.53 − 0.983i)31-s + (0.841 + 0.459i)33-s + (0.0779 − 0.0228i)36-s + (−1.50 − 0.559i)37-s + ⋯
L(s)  = 1  + (−0.936 + 0.203i)3-s + (−0.989 − 0.142i)4-s + (−0.909 + 0.415i)5-s + (−0.0739 + 0.0337i)9-s + (−0.755 − 0.654i)11-s + (0.956 − 0.0683i)12-s + (0.767 − 0.574i)15-s + (0.959 + 0.281i)16-s + (0.959 − 0.281i)20-s + (0.755 + 0.654i)23-s + (0.654 − 0.755i)25-s + (0.829 − 0.621i)27-s + (1.53 − 0.983i)31-s + (0.841 + 0.459i)33-s + (0.0779 − 0.0228i)36-s + (−1.50 − 0.559i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 + 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 + 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1265\)    =    \(5 \cdot 11 \cdot 23\)
Sign: $0.912 + 0.408i$
Analytic conductor: \(0.631317\)
Root analytic conductor: \(0.794554\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1265} (1022, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1265,\ (\ :0),\ 0.912 + 0.408i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3881305481\)
\(L(\frac12)\) \(\approx\) \(0.3881305481\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.909 - 0.415i)T \)
11 \( 1 + (0.755 + 0.654i)T \)
23 \( 1 + (-0.755 - 0.654i)T \)
good2 \( 1 + (0.989 + 0.142i)T^{2} \)
3 \( 1 + (0.936 - 0.203i)T + (0.909 - 0.415i)T^{2} \)
7 \( 1 + (-0.540 + 0.841i)T^{2} \)
13 \( 1 + (0.540 + 0.841i)T^{2} \)
17 \( 1 + (-0.281 - 0.959i)T^{2} \)
19 \( 1 + (0.959 + 0.281i)T^{2} \)
29 \( 1 + (-0.959 + 0.281i)T^{2} \)
31 \( 1 + (-1.53 + 0.983i)T + (0.415 - 0.909i)T^{2} \)
37 \( 1 + (1.50 + 0.559i)T + (0.755 + 0.654i)T^{2} \)
41 \( 1 + (0.654 + 0.755i)T^{2} \)
43 \( 1 + (0.909 - 0.415i)T^{2} \)
47 \( 1 + (-1.32 - 1.32i)T + iT^{2} \)
53 \( 1 + (-0.574 - 1.05i)T + (-0.540 + 0.841i)T^{2} \)
59 \( 1 + (0.557 + 1.89i)T + (-0.841 + 0.540i)T^{2} \)
61 \( 1 + (0.415 - 0.909i)T^{2} \)
67 \( 1 + (-1.75 - 0.125i)T + (0.989 + 0.142i)T^{2} \)
71 \( 1 + (0.857 + 0.989i)T + (-0.142 + 0.989i)T^{2} \)
73 \( 1 + (0.281 - 0.959i)T^{2} \)
79 \( 1 + (-0.841 + 0.540i)T^{2} \)
83 \( 1 + (0.755 + 0.654i)T^{2} \)
89 \( 1 + (0.474 + 0.304i)T + (0.415 + 0.909i)T^{2} \)
97 \( 1 + (-0.244 - 0.654i)T + (-0.755 + 0.654i)T^{2} \)
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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

−10.02039363024235885578794676634, −8.939645420029074809402543532487, −8.225891191894807336264651612893, −7.49053772272056058236125139616, −6.32647398565632856522360369746, −5.49348645093700719851018738725, −4.82024908586162874916427125656, −3.89811318805889444639023753691, −2.87403214108665766999493579470, −0.57367391216193611081334421548, 0.878367033183943946794112739360, 2.95823002748259960676611332538, 4.12799671148940120699341544134, 4.95857196376308317631574764992, 5.41765869801101297499268324798, 6.71364032677429804139491024676, 7.46790992027157223434650811170, 8.526765658200122649335988954430, 8.813661533155422074552813776747, 10.12266022413981027256594101520

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