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Properties

Label 2-315-9.7-c1-0-21
Degree $2$
Conductor $315$
Sign $-0.804 + 0.593i$
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  + (1.14 − 1.97i)2-s + (1.36 − 1.07i)3-s + (−1.60 − 2.78i)4-s + (−0.5 − 0.866i)5-s + (−0.565 − 3.91i)6-s + (−0.5 + 0.866i)7-s − 2.77·8-s + (0.704 − 2.91i)9-s − 2.28·10-s + (−1.98 + 3.44i)11-s + (−5.17 − 2.06i)12-s + (1.96 + 3.40i)13-s + (1.14 + 1.97i)14-s + (−1.60 − 0.642i)15-s + (0.0432 − 0.0748i)16-s + 7.02·17-s + ⋯
L(s)  = 1  + (0.807 − 1.39i)2-s + (0.785 − 0.618i)3-s + (−0.804 − 1.39i)4-s + (−0.223 − 0.387i)5-s + (−0.230 − 1.59i)6-s + (−0.188 + 0.327i)7-s − 0.982·8-s + (0.234 − 0.972i)9-s − 0.722·10-s + (−0.599 + 1.03i)11-s + (−1.49 − 0.596i)12-s + (0.545 + 0.944i)13-s + (0.305 + 0.528i)14-s + (−0.415 − 0.165i)15-s + (0.0108 − 0.0187i)16-s + 1.70·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.804 + 0.593i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.804 + 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.804 + 0.593i$
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: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.804 + 0.593i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.714988 - 2.17307i\)
\(L(\frac12)\) \(\approx\) \(0.714988 - 2.17307i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.36 + 1.07i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-1.14 + 1.97i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (1.98 - 3.44i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.96 - 3.40i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 7.02T + 17T^{2} \)
19 \( 1 + 7.41T + 19T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.36 + 5.82i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.52 - 2.64i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 7.65T + 37T^{2} \)
41 \( 1 + (-4.77 - 8.27i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.39 - 2.41i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.87 + 3.25i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 4.67T + 53T^{2} \)
59 \( 1 + (-5.76 - 9.98i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.03 + 1.78i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.96 + 5.13i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.91T + 71T^{2} \)
73 \( 1 + 5.52T + 73T^{2} \)
79 \( 1 + (-4.22 + 7.31i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.68 - 4.64i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 10.9T + 89T^{2} \)
97 \( 1 + (3.41 - 5.91i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.75291628040554422193977726441, −10.40900212247716842826619967080, −9.668122970099117182191824977939, −8.613707867365218009230230633840, −7.57255655930292781131281796627, −6.21091880430365890820539415283, −4.74918294742523031266994844104, −3.78803107925775943887588863094, −2.59487869470260862752410531160, −1.52658411824027499994366474433, 3.12296758484726437979231592869, 3.87622406403670447901268413479, 5.17507740116259487247298735040, 6.03680105349227614781100396608, 7.26776840181474316810789368751, 8.134709574363864214406478228201, 8.652639720264896123073758464388, 10.33559725489293913034896107086, 10.74296017216613525194505674962, 12.54902800597550938113985984851

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