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Properties

Label 2-1295-7.4-c1-0-81
Degree $2$
Conductor $1295$
Sign $0.437 + 0.899i$
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  + (0.458 + 0.794i)2-s + (1.13 − 1.97i)3-s + (0.579 − 1.00i)4-s + (−0.5 − 0.866i)5-s + 2.08·6-s + (2.28 + 1.33i)7-s + 2.89·8-s + (−1.09 − 1.89i)9-s + (0.458 − 0.794i)10-s + (2.39 − 4.14i)11-s + (−1.31 − 2.28i)12-s − 3.72·13-s + (−0.0181 + 2.42i)14-s − 2.27·15-s + (0.171 + 0.296i)16-s + (0.222 − 0.384i)17-s + ⋯
L(s)  = 1  + (0.324 + 0.561i)2-s + (0.657 − 1.13i)3-s + (0.289 − 0.501i)4-s + (−0.223 − 0.387i)5-s + 0.852·6-s + (0.862 + 0.506i)7-s + 1.02·8-s + (−0.364 − 0.630i)9-s + (0.145 − 0.251i)10-s + (0.721 − 1.24i)11-s + (−0.380 − 0.659i)12-s − 1.03·13-s + (−0.00485 + 0.648i)14-s − 0.587·15-s + (0.0427 + 0.0741i)16-s + (0.0538 − 0.0933i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1295 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1295 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1295\)    =    \(5 \cdot 7 \cdot 37\)
Sign: $0.437 + 0.899i$
Analytic conductor: \(10.3406\)
Root analytic conductor: \(3.21568\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1295} (186, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1295,\ (\ :1/2),\ 0.437 + 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.016052306\)
\(L(\frac12)\) \(\approx\) \(3.016052306\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.28 - 1.33i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-0.458 - 0.794i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.13 + 1.97i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-2.39 + 4.14i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.72T + 13T^{2} \)
17 \( 1 + (-0.222 + 0.384i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.38 - 4.13i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.214 + 0.372i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.38T + 29T^{2} \)
31 \( 1 + (3.13 - 5.42i)T + (-15.5 - 26.8i)T^{2} \)
41 \( 1 + 0.522T + 41T^{2} \)
43 \( 1 + 7.86T + 43T^{2} \)
47 \( 1 + (0.276 + 0.478i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.59 - 2.75i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.46 - 9.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.19 + 7.26i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.709 + 1.22i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.65T + 71T^{2} \)
73 \( 1 + (-0.899 + 1.55i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.60 + 6.24i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.45T + 83T^{2} \)
89 \( 1 + (-7.15 - 12.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.70T + 97T^{2} \)
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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

−9.193362507833205415396737383908, −8.419110513106988742389912452212, −7.79520723879559923849368552470, −7.14891507486113299950658025239, −6.24778861995078247641824128448, −5.45070226386494353579207056342, −4.63718673514389221550482078914, −3.19669091241297700855951349475, −1.93655608400698845923260135245, −1.17611309924738958603194684142, 1.82549608689684279930848764417, 2.82773769532531414961712230216, 3.75756131893812863207370606763, 4.47921846101750551364835600519, 4.94465379967563773386364360390, 6.84113974039522351754500757833, 7.39757784056668629912057659740, 8.161031861099020524286180107694, 9.227148606420534201661166627659, 9.913459545913492385975554452008

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