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Properties

Label 2-315-9.7-c1-0-14
Degree $2$
Conductor $315$
Sign $-0.173 + 0.984i$
Analytic cond. $2.51528$
Root an. cond. $1.58596$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)2-s + (−1.5 − 0.866i)3-s + (−0.999 − 1.73i)4-s + (0.5 + 0.866i)5-s + (3 − 1.73i)6-s + (−0.5 + 0.866i)7-s + (1.5 + 2.59i)9-s − 1.99·10-s + (−1.5 + 2.59i)11-s + 3.46i·12-s + (−3 − 5.19i)13-s + (−0.999 − 1.73i)14-s − 1.73i·15-s + (1.99 − 3.46i)16-s − 2·17-s − 6·18-s + ⋯
L(s)  = 1  + (−0.707 + 1.22i)2-s + (−0.866 − 0.499i)3-s + (−0.499 − 0.866i)4-s + (0.223 + 0.387i)5-s + (1.22 − 0.707i)6-s + (−0.188 + 0.327i)7-s + (0.5 + 0.866i)9-s − 0.632·10-s + (−0.452 + 0.783i)11-s + 0.999i·12-s + (−0.832 − 1.44i)13-s + (−0.267 − 0.462i)14-s − 0.447i·15-s + (0.499 − 0.866i)16-s − 0.485·17-s − 1.41·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.173 + 0.984i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 + 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (1 - 1.73i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3 + 5.19i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (5 + 8.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + (-3 - 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2 - 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.5 - 6.06i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 4T + 53T^{2} \)
59 \( 1 + (7 + 12.1i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2 - 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1 - 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 9T + 71T^{2} \)
73 \( 1 - 13T + 73T^{2} \)
79 \( 1 + (8.5 - 14.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.5 - 11.2i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (-4.5 + 7.79i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.28790245124406219325834414461, −10.23381064561320180503101258441, −9.558411675696091668225550396574, −8.083729258950178862141271812890, −7.55761457986048697693512678669, −6.56357979723283916516423988025, −5.84238684194872165135560994307, −4.86355685640166565028373406273, −2.49789149299085538948674805055, 0, 1.77021526492042973258993541278, 3.47942312485617092946418405229, 4.68138555273270764845834952430, 5.91938480869255224945259338172, 7.08062239230232568816955140112, 8.757670347959424035274928629413, 9.327142254780200113141852011914, 10.28100658031774493075400371767, 10.85599475480889022452999525775

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