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Properties

Label 2-315-63.47-c1-0-6
Degree $2$
Conductor $315$
Sign $0.586 - 0.810i$
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  + (1.36 − 0.788i)2-s + (−1.67 + 0.432i)3-s + (0.243 − 0.421i)4-s − 5-s + (−1.94 + 1.91i)6-s + (0.569 + 2.58i)7-s + 2.38i·8-s + (2.62 − 1.45i)9-s + (−1.36 + 0.788i)10-s + 4.04i·11-s + (−0.225 + 0.811i)12-s + (2.45 − 1.41i)13-s + (2.81 + 3.07i)14-s + (1.67 − 0.432i)15-s + (2.36 + 4.10i)16-s + (2.76 + 4.79i)17-s + ⋯
L(s)  = 1  + (0.965 − 0.557i)2-s + (−0.968 + 0.249i)3-s + (0.121 − 0.210i)4-s − 0.447·5-s + (−0.795 + 0.781i)6-s + (0.215 + 0.976i)7-s + 0.843i·8-s + (0.875 − 0.483i)9-s + (−0.431 + 0.249i)10-s + 1.21i·11-s + (−0.0650 + 0.234i)12-s + (0.680 − 0.392i)13-s + (0.752 + 0.822i)14-s + (0.433 − 0.111i)15-s + (0.592 + 1.02i)16-s + (0.670 + 1.16i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.586 - 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.586 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.586 - 0.810i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (236, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.586 - 0.810i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21532 + 0.620687i\)
\(L(\frac12)\) \(\approx\) \(1.21532 + 0.620687i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.67 - 0.432i)T \)
5 \( 1 + T \)
7 \( 1 + (-0.569 - 2.58i)T \)
good2 \( 1 + (-1.36 + 0.788i)T + (1 - 1.73i)T^{2} \)
11 \( 1 - 4.04iT - 11T^{2} \)
13 \( 1 + (-2.45 + 1.41i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.76 - 4.79i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (7.37 + 4.25i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 0.953iT - 23T^{2} \)
29 \( 1 + (-2.77 - 1.60i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.04 + 1.18i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.10 + 1.90i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.62 + 2.80i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.84 + 4.93i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.16 - 7.21i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.73 + 2.73i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.74 + 6.48i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.757 + 0.437i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.90 + 6.76i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.34iT - 71T^{2} \)
73 \( 1 + (-14.1 + 8.18i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.50 + 6.07i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.79 - 4.83i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.18 + 2.06i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-14.7 - 8.51i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.02538263096549689974445704204, −11.04441319646955297238605553864, −10.47689566257714300426848970473, −9.022637890265299910101551189274, −8.030901077054702696562431192242, −6.55928099009092285298608808127, −5.55427522532643227021818100929, −4.64364658907551012558433064496, −3.79256179542093493558227121993, −2.13297141323560877277666612757, 0.877769943705403896899247443878, 3.69027905600655573797298691541, 4.49336571805372753370561482929, 5.62355091839086443498340118942, 6.42799720312052306799912778866, 7.27025022560095026668761874129, 8.359005631428340064316305809961, 9.972933240022811645171170532628, 10.81633646178853299102510317978, 11.60368551858452584130568506395

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