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Properties

Label 2-315-7.2-c1-0-5
Degree $2$
Conductor $315$
Sign $0.605 - 0.795i$
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 + 1.73i)4-s + (−0.5 + 0.866i)5-s + (2.5 − 0.866i)7-s − 13-s + (−1.99 + 3.46i)16-s + (3 + 5.19i)17-s + (−2.5 + 4.33i)19-s − 1.99·20-s + (3 − 5.19i)23-s + (−0.499 − 0.866i)25-s + (4 + 3.46i)28-s + 6·29-s + (−2.5 − 4.33i)31-s + (−0.500 + 2.59i)35-s + (3.5 − 6.06i)37-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)4-s + (−0.223 + 0.387i)5-s + (0.944 − 0.327i)7-s − 0.277·13-s + (−0.499 + 0.866i)16-s + (0.727 + 1.26i)17-s + (−0.573 + 0.993i)19-s − 0.447·20-s + (0.625 − 1.08i)23-s + (−0.0999 − 0.173i)25-s + (0.755 + 0.654i)28-s + 1.11·29-s + (−0.449 − 0.777i)31-s + (−0.0845 + 0.439i)35-s + (0.575 − 0.996i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34361 + 0.666068i\)
\(L(\frac12)\) \(\approx\) \(1.34361 + 0.666068i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-2.5 + 0.866i)T \)
good2 \( 1 + (-1 - 1.73i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 12T + 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (5.5 + 9.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.5 + 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−11.88914951292933934468641466781, −10.83262368802642757133536186580, −10.29697573743675917707079220577, −8.584220237126951316049148906100, −8.016420556772781537181595938986, −7.13266009033408784716826599938, −6.05381080214921419330021229560, −4.49085993190327801605516736560, −3.47559153022898873773018368587, −1.99930704427235173080527762837, 1.26808572845763836649430746814, 2.79339907232768841076749241642, 4.80731631987292312102069204479, 5.29361037150917152580677605651, 6.69386668812192113543142770088, 7.62568741502757608385075518049, 8.781759805368727545139118771297, 9.664568004548508713651560743111, 10.71032054739340183406463419753, 11.56435231601265826339740340824

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