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Properties

Label 2-315-63.4-c1-0-4
Degree $2$
Conductor $315$
Sign $-0.144 - 0.989i$
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  + (0.517 − 0.896i)2-s + (−1.26 + 1.18i)3-s + (0.463 + 0.803i)4-s − 5-s + (0.402 + 1.74i)6-s + (−1.07 + 2.41i)7-s + 3.03·8-s + (0.211 − 2.99i)9-s + (−0.517 + 0.896i)10-s − 5.83·11-s + (−1.53 − 0.470i)12-s + (−1.79 + 3.10i)13-s + (1.60 + 2.21i)14-s + (1.26 − 1.18i)15-s + (0.642 − 1.11i)16-s + (−1.70 + 2.96i)17-s + ⋯
L(s)  = 1  + (0.366 − 0.634i)2-s + (−0.731 + 0.681i)3-s + (0.231 + 0.401i)4-s − 0.447·5-s + (0.164 + 0.713i)6-s + (−0.407 + 0.913i)7-s + 1.07·8-s + (0.0705 − 0.997i)9-s + (−0.163 + 0.283i)10-s − 1.76·11-s + (−0.443 − 0.135i)12-s + (−0.497 + 0.861i)13-s + (0.430 + 0.592i)14-s + (0.327 − 0.304i)15-s + (0.160 − 0.278i)16-s + (−0.414 + 0.717i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.144 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.144 - 0.989i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.144 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.602210 + 0.696374i\)
\(L(\frac12)\) \(\approx\) \(0.602210 + 0.696374i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.26 - 1.18i)T \)
5 \( 1 + T \)
7 \( 1 + (1.07 - 2.41i)T \)
good2 \( 1 + (-0.517 + 0.896i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + 5.83T + 11T^{2} \)
13 \( 1 + (1.79 - 3.10i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.70 - 2.96i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.47 - 4.28i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 6.56T + 23T^{2} \)
29 \( 1 + (1.65 + 2.86i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.07 + 1.86i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.0816 - 0.141i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.52 + 7.83i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.873 - 1.51i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.56 - 6.17i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.06 + 5.31i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.59 - 2.76i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.41 - 4.18i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.13 - 14.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.78T + 71T^{2} \)
73 \( 1 + (-5.42 + 9.39i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.53 + 6.12i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.158 - 0.274i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.01 - 1.75i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.865 + 1.49i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.87764165971014639598955256041, −11.10486172162103016840914882137, −10.38316843353250693543851110463, −9.349874946402199339665692385850, −8.136251159424027993811947509924, −7.05296679270226521229730327316, −5.70453556996868657592357339518, −4.75677004430098070835197217750, −3.60749670513368056617658829815, −2.44514626735733334778361031342, 0.62407767021724644380957046299, 2.77885217348333193688619056604, 4.90593793701217992899703555398, 5.27064291068582752015045498348, 6.71601037429758042267547524187, 7.30984819572704992776158676635, 7.941988071189493103367374832411, 9.799481193493693128204758353757, 10.87728996531041273305951485042, 11.05154360384709036732054579294

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