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Properties

Label 2-1295-7.2-c1-0-62
Degree $2$
Conductor $1295$
Sign $-0.229 + 0.973i$
Analytic cond. $10.3406$
Root an. cond. $3.21568$
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  + (1.20 − 2.09i)2-s + (0.744 + 1.28i)3-s + (−1.92 − 3.33i)4-s + (−0.5 + 0.866i)5-s + 3.59·6-s + (−1.93 + 1.80i)7-s − 4.46·8-s + (0.392 − 0.679i)9-s + (1.20 + 2.09i)10-s + (−1.94 − 3.37i)11-s + (2.86 − 4.96i)12-s + 6.09·13-s + (1.45 + 6.23i)14-s − 1.48·15-s + (−1.55 + 2.69i)16-s + (1.95 + 3.38i)17-s + ⋯
L(s)  = 1  + (0.854 − 1.48i)2-s + (0.429 + 0.744i)3-s + (−0.962 − 1.66i)4-s + (−0.223 + 0.387i)5-s + 1.46·6-s + (−0.729 + 0.683i)7-s − 1.58·8-s + (0.130 − 0.226i)9-s + (0.382 + 0.662i)10-s + (−0.586 − 1.01i)11-s + (0.826 − 1.43i)12-s + 1.69·13-s + (0.388 + 1.66i)14-s − 0.384·15-s + (−0.388 + 0.673i)16-s + (0.474 + 0.821i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1295 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1295 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1295\)    =    \(5 \cdot 7 \cdot 37\)
Sign: $-0.229 + 0.973i$
Analytic conductor: \(10.3406\)
Root analytic conductor: \(3.21568\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1295} (926, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1295,\ (\ :1/2),\ -0.229 + 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.748357626\)
\(L(\frac12)\) \(\approx\) \(2.748357626\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (1.93 - 1.80i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-1.20 + 2.09i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.744 - 1.28i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (1.94 + 3.37i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 6.09T + 13T^{2} \)
17 \( 1 + (-1.95 - 3.38i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.37 + 5.83i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.36 + 4.10i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.37T + 29T^{2} \)
31 \( 1 + (0.687 + 1.18i)T + (-15.5 + 26.8i)T^{2} \)
41 \( 1 + 2.65T + 41T^{2} \)
43 \( 1 - 3.20T + 43T^{2} \)
47 \( 1 + (6.26 - 10.8i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.23 - 2.13i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.871 + 1.50i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.50 + 7.80i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.07 - 3.60i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.11T + 71T^{2} \)
73 \( 1 + (-1.74 - 3.02i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.44 + 9.43i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 8.09T + 83T^{2} \)
89 \( 1 + (2.52 - 4.37i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.5T + 97T^{2} \)
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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

−9.595104979593029244551355177778, −8.987516398686728159219344508814, −8.209583382850193976278151313617, −6.55525765877608315969027012587, −5.83451267581171162321592816672, −4.84109466813143983109174978078, −3.76685616643070266336750990642, −3.26248237103764099725064246979, −2.67709665423594617478679149268, −0.966985398822360081550444989345, 1.40765281299409748360962807735, 3.24019786447130454782588933868, 3.96367551705224386722063855012, 5.03336215647822826450611367899, 5.77225468036534123907648312057, 6.80933269884179574334926104296, 7.30222382737993333264481733175, 7.905778788011836560869320549808, 8.565869306587437867995685818751, 9.680678495308676978962037046781

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