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Properties

Label 2-1274-91.25-c1-0-28
Degree $2$
Conductor $1274$
Sign $0.926 + 0.377i$
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  + (0.866 − 0.5i)2-s + (−0.207 + 0.358i)3-s + (0.499 − 0.866i)4-s + (1.22 − 0.707i)5-s + 0.414i·6-s − 0.999i·8-s + (1.41 + 2.44i)9-s + (0.707 − 1.22i)10-s + (2.80 + 1.62i)11-s + (0.207 + 0.358i)12-s + (3 − 2i)13-s + 0.585i·15-s + (−0.5 − 0.866i)16-s + (3 − 5.19i)17-s + (2.44 + 1.41i)18-s + (−7.13 + 4.12i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.119 + 0.207i)3-s + (0.249 − 0.433i)4-s + (0.547 − 0.316i)5-s + 0.169i·6-s − 0.353i·8-s + (0.471 + 0.816i)9-s + (0.223 − 0.387i)10-s + (0.846 + 0.488i)11-s + (0.0597 + 0.103i)12-s + (0.832 − 0.554i)13-s + 0.151i·15-s + (−0.125 − 0.216i)16-s + (0.727 − 1.26i)17-s + (0.577 + 0.333i)18-s + (−1.63 + 0.945i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.377i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.926 + 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1274\)    =    \(2 \cdot 7^{2} \cdot 13\)
Sign: $0.926 + 0.377i$
Analytic conductor: \(10.1729\)
Root analytic conductor: \(3.18950\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1274} (753, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1274,\ (\ :1/2),\ 0.926 + 0.377i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.791694811\)
\(L(\frac12)\) \(\approx\) \(2.791694811\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 \)
13 \( 1 + (-3 + 2i)T \)
good3 \( 1 + (0.207 - 0.358i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.22 + 0.707i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.80 - 1.62i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (7.13 - 4.12i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.20 - 3.82i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.58T + 29T^{2} \)
31 \( 1 + (-2.80 - 1.62i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-7.49 + 4.32i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 9iT - 41T^{2} \)
43 \( 1 - 12.2T + 43T^{2} \)
47 \( 1 + (1.37 - 0.792i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.707 + 1.22i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.52 - 0.878i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.5 - 2.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.97 + 3.44i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.2iT - 71T^{2} \)
73 \( 1 + (3.01 + 1.74i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.86 - 11.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.41iT - 83T^{2} \)
89 \( 1 + (9.79 - 5.65i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 7.48iT - 97T^{2} \)
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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

−9.643738830738779206040180109316, −9.133315944929487559758563723697, −7.893199965959121821887910820308, −7.11673338445026432891267621782, −5.94744713461343379236010827898, −5.44626540697823235540066886095, −4.40679135183799979851369712192, −3.67450252589541431697189939056, −2.28362669221746494553466677042, −1.30560393672778075701108290535, 1.27439380469876428388181039276, 2.59621046696196246266253291575, 3.89452531040063567596275718942, 4.36298732709136484220483532020, 6.01432660665299783961074600929, 6.23689282681935814929703848392, 6.85661031180242101048985822909, 8.100239397644269599896650754404, 8.837694602079914956693247335728, 9.647064912048955203532807515629

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