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Properties

Label 2-1295-7.4-c1-0-78
Degree $2$
Conductor $1295$
Sign $0.995 + 0.0894i$
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  + (1.04 + 1.80i)2-s + (1.31 − 2.27i)3-s + (−1.18 + 2.04i)4-s + (−0.5 − 0.866i)5-s + 5.49·6-s + (2.32 − 1.26i)7-s − 0.766·8-s + (−1.95 − 3.38i)9-s + (1.04 − 1.80i)10-s + (−0.0389 + 0.0673i)11-s + (3.11 + 5.38i)12-s + 1.91·13-s + (4.71 + 2.88i)14-s − 2.62·15-s + (1.56 + 2.71i)16-s + (−0.652 + 1.13i)17-s + ⋯
L(s)  = 1  + (0.738 + 1.27i)2-s + (0.759 − 1.31i)3-s + (−0.591 + 1.02i)4-s + (−0.223 − 0.387i)5-s + 2.24·6-s + (0.878 − 0.477i)7-s − 0.271·8-s + (−0.652 − 1.12i)9-s + (0.330 − 0.572i)10-s + (−0.0117 + 0.0203i)11-s + (0.898 + 1.55i)12-s + 0.529·13-s + (1.25 + 0.772i)14-s − 0.678·15-s + (0.391 + 0.677i)16-s + (−0.158 + 0.274i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1295 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1295 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1295\)    =    \(5 \cdot 7 \cdot 37\)
Sign: $0.995 + 0.0894i$
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: \((2,\ 1295,\ (\ :1/2),\ 0.995 + 0.0894i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.414873916\)
\(L(\frac12)\) \(\approx\) \(3.414873916\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.32 + 1.26i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-1.04 - 1.80i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.31 + 2.27i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (0.0389 - 0.0673i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.91T + 13T^{2} \)
17 \( 1 + (0.652 - 1.13i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.61 + 2.79i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.83 + 3.18i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.75T + 29T^{2} \)
31 \( 1 + (-1.55 + 2.69i)T + (-15.5 - 26.8i)T^{2} \)
41 \( 1 - 6.69T + 41T^{2} \)
43 \( 1 - 2.18T + 43T^{2} \)
47 \( 1 + (-2.06 - 3.57i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.77 - 3.07i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.74 + 3.03i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.74 - 8.22i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.815 - 1.41i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.65T + 71T^{2} \)
73 \( 1 + (2.70 - 4.68i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.10 + 1.91i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.20T + 83T^{2} \)
89 \( 1 + (-4.60 - 7.97i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 6.48T + 97T^{2} \)
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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

−9.024381774274117763297029325115, −8.399559039755583974508184998964, −7.76221209802506832323705689585, −7.29539603197051262309908583470, −6.47299411747283701810428792172, −5.68566493469691746971574898057, −4.57740946611046695375823779987, −3.87283513550850988847340857390, −2.32250924812879779094890230840, −1.16324384788356279827675491268, 1.74721471254801402164576688570, 2.71839175751791636377505855606, 3.59787882940937963647790799146, 4.17126241142950955611723474281, 4.98400751642589158066747421528, 5.84714051558077227800307754071, 7.47365698332453447536734759016, 8.326903915370736015784027863066, 9.123969764083818330919221261115, 9.901900989902946171893257906848

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