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Properties

Label 2-315-63.16-c1-0-31
Degree $2$
Conductor $315$
Sign $-0.526 - 0.850i$
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  + (−0.750 − 1.29i)2-s + (−0.590 − 1.62i)3-s + (−0.125 + 0.218i)4-s + 5-s + (−1.67 + 1.98i)6-s + (−2.32 + 1.25i)7-s − 2.62·8-s + (−2.30 + 1.92i)9-s + (−0.750 − 1.29i)10-s − 2.83·11-s + (0.429 + 0.0763i)12-s + (0.0336 + 0.0582i)13-s + (3.37 + 2.08i)14-s + (−0.590 − 1.62i)15-s + (2.22 + 3.84i)16-s + (−3.48 − 6.02i)17-s + ⋯
L(s)  = 1  + (−0.530 − 0.918i)2-s + (−0.340 − 0.940i)3-s + (−0.0629 + 0.109i)4-s + 0.447·5-s + (−0.683 + 0.811i)6-s + (−0.880 + 0.473i)7-s − 0.927·8-s + (−0.767 + 0.640i)9-s + (−0.237 − 0.410i)10-s − 0.853·11-s + (0.123 + 0.0220i)12-s + (0.00932 + 0.0161i)13-s + (0.902 + 0.557i)14-s + (−0.152 − 0.420i)15-s + (0.555 + 0.961i)16-s + (−0.844 − 1.46i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.526 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.526 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.526 - 0.850i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.526 - 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.189205 + 0.339623i\)
\(L(\frac12)\) \(\approx\) \(0.189205 + 0.339623i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.590 + 1.62i)T \)
5 \( 1 - T \)
7 \( 1 + (2.32 - 1.25i)T \)
good2 \( 1 + (0.750 + 1.29i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + 2.83T + 11T^{2} \)
13 \( 1 + (-0.0336 - 0.0582i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.48 + 6.02i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.667 + 1.15i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 3.22T + 23T^{2} \)
29 \( 1 + (-4.51 + 7.81i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.73 - 8.20i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.37 + 2.38i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.36 - 4.09i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.96 + 6.86i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.867 + 1.50i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.77 + 11.7i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.43 + 5.95i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.89 + 3.28i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.82 - 3.16i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.89T + 71T^{2} \)
73 \( 1 + (-2.58 - 4.46i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.73 + 9.93i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.40 - 9.36i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.42 - 5.93i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.45 + 2.52i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.13215619977947360474446610855, −10.16275741652310546690539832006, −9.363721437280737097775256887978, −8.429051231977840743849658638050, −7.05657249844500176175897030854, −6.18384935862064997820906179715, −5.23097217610292805829482443808, −2.91332645505349568103568278411, −2.16105307849086907783383853371, −0.31666620932446391877684032505, 2.96034924471688919795436715970, 4.23481413143882835625187858824, 5.78695136729853988864084915906, 6.27916810749981954391082672914, 7.49317168127166525890132353203, 8.589040702394915703010974585130, 9.404806132149889360257349700064, 10.26803339791027095377076926734, 10.96374616383240515796435725701, 12.34200381679903053751763448682

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