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Properties

Label 2-315-105.89-c1-0-12
Degree $2$
Conductor $315$
Sign $-0.0197 + 0.999i$
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  + (0.956 − 1.65i)2-s + (−0.830 − 1.43i)4-s + (2.17 + 0.526i)5-s + (−1.11 − 2.39i)7-s + 0.650·8-s + (2.95 − 3.09i)10-s + (−2.79 + 1.61i)11-s + 4.86·13-s + (−5.04 − 0.439i)14-s + (2.28 − 3.95i)16-s + (0.631 − 0.364i)17-s + (−6.81 − 3.93i)19-s + (−1.04 − 3.56i)20-s + 6.17i·22-s + (−2.43 + 4.21i)23-s + ⋯
L(s)  = 1  + (0.676 − 1.17i)2-s + (−0.415 − 0.718i)4-s + (0.971 + 0.235i)5-s + (−0.422 − 0.906i)7-s + 0.229·8-s + (0.933 − 0.979i)10-s + (−0.842 + 0.486i)11-s + 1.34·13-s + (−1.34 − 0.117i)14-s + (0.570 − 0.988i)16-s + (0.153 − 0.0884i)17-s + (−1.56 − 0.902i)19-s + (−0.234 − 0.796i)20-s + 1.31i·22-s + (−0.507 + 0.879i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0197 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0197 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.0197 + 0.999i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.0197 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45170 - 1.48065i\)
\(L(\frac12)\) \(\approx\) \(1.45170 - 1.48065i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-2.17 - 0.526i)T \)
7 \( 1 + (1.11 + 2.39i)T \)
good2 \( 1 + (-0.956 + 1.65i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (2.79 - 1.61i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.86T + 13T^{2} \)
17 \( 1 + (-0.631 + 0.364i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.81 + 3.93i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.43 - 4.21i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.75iT - 29T^{2} \)
31 \( 1 + (1.23 - 0.714i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.74 + 1.58i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.40T + 41T^{2} \)
43 \( 1 - 6.42iT - 43T^{2} \)
47 \( 1 + (-4.21 - 2.43i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.760 - 1.31i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.15 - 5.45i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.05 + 1.18i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.63 + 5.56i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.1iT - 71T^{2} \)
73 \( 1 + (6.91 + 11.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.99 - 3.44i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.19iT - 83T^{2} \)
89 \( 1 + (-5.63 + 9.76i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2.21T + 97T^{2} \)
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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

−11.18284981885659461660370047680, −10.64774345523484478977378256566, −10.06211453165158485066861661514, −8.910842215734958932790950312195, −7.44269473507319003834028291436, −6.37682033685997001632823805616, −5.13508783714197293199681780418, −3.97345701074463450974199139827, −2.87606741865795254324198312577, −1.57896325303136057946758594972, 2.18575284212352528117785277170, 3.95201052215028105511450368186, 5.35894658307984785960842382070, 6.01291214552258039353966366116, 6.52351711743757532935404234175, 8.210182363460170633861429760396, 8.633544098276879309084888246496, 10.06013660617471333148007478062, 10.80417982557768073264281555854, 12.32674224571106019967031030739

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