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Properties

Label 2-315-63.4-c1-0-6
Degree $2$
Conductor $315$
Sign $-0.821 - 0.570i$
Analytic cond. $2.51528$
Root an. cond. $1.58596$
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  + (−0.148 + 0.257i)2-s + (0.310 + 1.70i)3-s + (0.955 + 1.65i)4-s − 5-s + (−0.485 − 0.173i)6-s + (−2.64 − 0.0857i)7-s − 1.16·8-s + (−2.80 + 1.05i)9-s + (0.148 − 0.257i)10-s + 4.84·11-s + (−2.52 + 2.14i)12-s + (−2.59 + 4.49i)13-s + (0.415 − 0.668i)14-s + (−0.310 − 1.70i)15-s + (−1.73 + 3.01i)16-s + (1.80 − 3.11i)17-s + ⋯
L(s)  = 1  + (−0.105 + 0.182i)2-s + (0.179 + 0.983i)3-s + (0.477 + 0.827i)4-s − 0.447·5-s + (−0.198 − 0.0707i)6-s + (−0.999 − 0.0324i)7-s − 0.411·8-s + (−0.935 + 0.353i)9-s + (0.0470 − 0.0814i)10-s + 1.46·11-s + (−0.728 + 0.618i)12-s + (−0.719 + 1.24i)13-s + (0.111 − 0.178i)14-s + (−0.0802 − 0.439i)15-s + (−0.434 + 0.752i)16-s + (0.436 − 0.756i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.821 - 0.570i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.821 - 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.821 - 0.570i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.821 - 0.570i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.327493 + 1.04456i\)
\(L(\frac12)\) \(\approx\) \(0.327493 + 1.04456i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.310 - 1.70i)T \)
5 \( 1 + T \)
7 \( 1 + (2.64 + 0.0857i)T \)
good2 \( 1 + (0.148 - 0.257i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 - 4.84T + 11T^{2} \)
13 \( 1 + (2.59 - 4.49i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.80 + 3.11i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.03 + 3.51i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 0.983T + 23T^{2} \)
29 \( 1 + (-2.30 - 3.98i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.60 - 6.23i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.25 - 5.63i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.298 - 0.516i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.0565 - 0.0979i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.57 - 6.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.78 + 8.29i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.75 - 6.49i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.49 + 12.9i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.78 - 6.55i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.2T + 71T^{2} \)
73 \( 1 + (1.67 - 2.90i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.59 - 4.49i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.45 + 9.45i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.64 + 13.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.85 - 11.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.77567179925229645485246789163, −11.37131006819159391097603270962, −9.956617334405983484116081746138, −9.181159501033769489045573908417, −8.508646847140913161443479345979, −7.02108749585440974368298738827, −6.53891235071652500996475173641, −4.71059909770432977902921753989, −3.73730456748897050130633203833, −2.79016273355352793584093706263, 0.802136450532578940710193037975, 2.42447528801500510678806094678, 3.72922188185437089919203994218, 5.74997578482884952165307112929, 6.36562116234720203143451448284, 7.30419158765127314124773820985, 8.375795183044322007205343546722, 9.534331517667855329623470246018, 10.29335486208915066197638454790, 11.48256632371641572258355216612

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