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Properties

Label 2-1274-91.23-c1-0-30
Degree $2$
Conductor $1274$
Sign $-0.451 + 0.892i$
Analytic cond. $10.1729$
Root an. cond. $3.18950$
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  − i·2-s + (0.126 − 0.218i)3-s − 4-s + (−0.993 − 0.573i)5-s + (−0.218 − 0.126i)6-s + i·8-s + (1.46 + 2.54i)9-s + (−0.573 + 0.993i)10-s + (−3.84 − 2.22i)11-s + (−0.126 + 0.218i)12-s + (3.54 + 0.666i)13-s + (−0.251 + 0.145i)15-s + 16-s + 2.70·17-s + (2.54 − 1.46i)18-s + (5.68 − 3.28i)19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.0729 − 0.126i)3-s − 0.5·4-s + (−0.444 − 0.256i)5-s + (−0.0894 − 0.0516i)6-s + 0.353i·8-s + (0.489 + 0.847i)9-s + (−0.181 + 0.314i)10-s + (−1.16 − 0.670i)11-s + (−0.0364 + 0.0632i)12-s + (0.982 + 0.184i)13-s + (−0.0649 + 0.0374i)15-s + 0.250·16-s + 0.657·17-s + (0.599 − 0.346i)18-s + (1.30 − 0.752i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1274\)    =    \(2 \cdot 7^{2} \cdot 13\)
Sign: $-0.451 + 0.892i$
Analytic conductor: \(10.1729\)
Root analytic conductor: \(3.18950\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1274} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1274,\ (\ :1/2),\ -0.451 + 0.892i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.340450899\)
\(L(\frac12)\) \(\approx\) \(1.340450899\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
7 \( 1 \)
13 \( 1 + (-3.54 - 0.666i)T \)
good3 \( 1 + (-0.126 + 0.218i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.993 + 0.573i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.84 + 2.22i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 2.70T + 17T^{2} \)
19 \( 1 + (-5.68 + 3.28i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 2.07T + 23T^{2} \)
29 \( 1 + (3.59 + 6.23i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (6.84 - 3.95i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 9.64iT - 37T^{2} \)
41 \( 1 + (-8.22 + 4.74i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.70 - 2.94i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.45 + 0.837i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.64 + 11.5i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 0.0677iT - 59T^{2} \)
61 \( 1 + (-4.05 - 7.01i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.444 - 0.256i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-9.34 - 5.39i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-6.94 + 4.01i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.37 - 9.30i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 15.3iT - 83T^{2} \)
89 \( 1 + 10.8iT - 89T^{2} \)
97 \( 1 + (1.84 + 1.06i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.492688030760730678832275017687, −8.597963226226835061932881615058, −7.85926377529258167854526548225, −7.27559173963763575534937461360, −5.75236463200111624101197863874, −5.14622554314530041578627450678, −4.05618373256305048920518479165, −3.15935318737305502900880813887, −2.03900099463108320780107307601, −0.63949987250342124289388078099, 1.28193880031822577229445371848, 3.19709866321726411287218768455, 3.80458141648701171840126996053, 5.03469673002116020264312718500, 5.74529469201696448314693272108, 6.75952758330136319678733199520, 7.62588043971862740641368140945, 7.959135393588326229650840752267, 9.231982827065153480837804633179, 9.696943329022719463809061058159

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