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Properties

Label 2-1264-316.103-c1-0-5
Degree $2$
Conductor $1264$
Sign $-0.852 + 0.522i$
Analytic cond. $10.0930$
Root an. cond. $3.17696$
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  + (−1.29 + 2.25i)3-s + (2.04 + 3.54i)5-s + (−1.22 − 2.11i)7-s + (−1.87 − 3.25i)9-s + (3.04 − 1.75i)11-s + (−2.87 + 4.98i)13-s − 10.6·15-s − 4.08i·17-s + (−2.63 + 1.52i)19-s + 6.36·21-s + (−2.45 + 1.41i)23-s + (−5.89 + 10.2i)25-s + 1.97·27-s + (−4.98 + 2.87i)29-s + (−6.87 + 3.96i)31-s + ⋯
L(s)  = 1  + (−0.750 + 1.29i)3-s + (0.916 + 1.58i)5-s + (−0.462 − 0.801i)7-s + (−0.626 − 1.08i)9-s + (0.917 − 0.529i)11-s + (−0.798 + 1.38i)13-s − 2.75·15-s − 0.991i·17-s + (−0.604 + 0.349i)19-s + 1.38·21-s + (−0.512 + 0.295i)23-s + (−1.17 + 2.04i)25-s + 0.379·27-s + (−0.925 + 0.534i)29-s + (−1.23 + 0.712i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.852 + 0.522i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1264\)    =    \(2^{4} \cdot 79\)
Sign: $-0.852 + 0.522i$
Analytic conductor: \(10.0930\)
Root analytic conductor: \(3.17696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1264} (735, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1264,\ (\ :1/2),\ -0.852 + 0.522i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7917958778\)
\(L(\frac12)\) \(\approx\) \(0.7917958778\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
79 \( 1 + (-6.82 - 5.69i)T \)
good3 \( 1 + (1.29 - 2.25i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2.04 - 3.54i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.22 + 2.11i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3.04 + 1.75i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.87 - 4.98i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.08iT - 17T^{2} \)
19 \( 1 + (2.63 - 1.52i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.45 - 1.41i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.98 - 2.87i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.87 - 3.96i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.34 - 1.92i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.62iT - 41T^{2} \)
43 \( 1 + (-1.64 + 2.85i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.01 + 8.68i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.86 + 2.80i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.60 - 7.98i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 5.98iT - 61T^{2} \)
67 \( 1 + 9.57iT - 67T^{2} \)
71 \( 1 - 15.3T + 71T^{2} \)
73 \( 1 + (3.77 + 6.53i)T + (-36.5 + 63.2i)T^{2} \)
83 \( 1 + (13.8 - 7.97i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 12.5T + 89T^{2} \)
97 \( 1 + 5.62T + 97T^{2} \)
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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

−10.12933175302428429479894942480, −9.613788251749968594376330144100, −9.090909995684860565130601334974, −7.25025261342656544149476283459, −6.78910263509616353711199974790, −6.04834051676492052116651178732, −5.14492197982083782402545118620, −3.97494646034379069796396509731, −3.44158417891337060467174007426, −2.05034018892328497504659504626, 0.34924957895243603083001212586, 1.58683750333522271533273821270, 2.32470147914911330213196705173, 4.22602668659971055949348141260, 5.36984160589090135817414757814, 5.85852604014124994483936749008, 6.42736925818514076999077462780, 7.62440720740982286470380172747, 8.314956967488629505498177248068, 9.302357808889825140404852425006

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