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Properties

Label 2-1260-105.23-c1-0-0
Degree $2$
Conductor $1260$
Sign $-0.228 - 0.973i$
Analytic cond. $10.0611$
Root an. cond. $3.17193$
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  + (−0.224 − 2.22i)5-s + (−2.29 − 1.31i)7-s + (−5.56 + 3.21i)11-s + (1.53 + 1.53i)13-s + (0.985 − 3.67i)17-s + (1.19 + 0.687i)19-s + (2.18 + 8.15i)23-s + (−4.89 + 0.997i)25-s − 5.95·29-s + (4.97 + 8.61i)31-s + (−2.40 + 5.40i)35-s + (0.642 + 2.39i)37-s − 1.18i·41-s + (3.28 + 3.28i)43-s + (−2.44 + 0.655i)47-s + ⋯
L(s)  = 1  + (−0.100 − 0.994i)5-s + (−0.868 − 0.495i)7-s + (−1.67 + 0.967i)11-s + (0.425 + 0.425i)13-s + (0.239 − 0.892i)17-s + (0.273 + 0.157i)19-s + (0.455 + 1.70i)23-s + (−0.979 + 0.199i)25-s − 1.10·29-s + (0.893 + 1.54i)31-s + (−0.406 + 0.913i)35-s + (0.105 + 0.394i)37-s − 0.184i·41-s + (0.500 + 0.500i)43-s + (−0.356 + 0.0955i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.228 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.228 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.228 - 0.973i$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ -0.228 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5381462424\)
\(L(\frac12)\) \(\approx\) \(0.5381462424\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (0.224 + 2.22i)T \)
7 \( 1 + (2.29 + 1.31i)T \)
good11 \( 1 + (5.56 - 3.21i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.53 - 1.53i)T + 13iT^{2} \)
17 \( 1 + (-0.985 + 3.67i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.19 - 0.687i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.18 - 8.15i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 5.95T + 29T^{2} \)
31 \( 1 + (-4.97 - 8.61i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.642 - 2.39i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 1.18iT - 41T^{2} \)
43 \( 1 + (-3.28 - 3.28i)T + 43iT^{2} \)
47 \( 1 + (2.44 - 0.655i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.80 + 0.482i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (3.27 + 5.67i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.09 + 7.08i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.29 - 2.49i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 8.06iT - 71T^{2} \)
73 \( 1 + (3.47 - 12.9i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (8.00 + 4.62i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.11 - 5.11i)T - 83iT^{2} \)
89 \( 1 + (3.97 - 6.88i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.537 - 0.537i)T - 97iT^{2} \)
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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

−9.618657684403094424072976938394, −9.451173651361301129648348268573, −8.180401359299898101843152241577, −7.53242969276834713702783381677, −6.77577367333175830804536473476, −5.45019627475854728822212075220, −4.99218090592220674641636742256, −3.86735265344670567665646920123, −2.82379587197214224154125146217, −1.35959329213938450299442667566, 0.22814472922600988344568554378, 2.51134768039578055129387726887, 3.00136381490060794883239300106, 4.06073751086983325782994228035, 5.61078563540720065636102168672, 5.97180384194487339383292831594, 6.93438379041756751296940711005, 7.931181629984004212892392407924, 8.472091166197167245342194336573, 9.589241306738854527900134394134

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