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Properties

Label 2-315-315.194-c1-0-34
Degree $2$
Conductor $315$
Sign $-0.986 + 0.163i$
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  − 0.932·2-s + (−0.704 − 1.58i)3-s − 1.13·4-s + (1.20 − 1.88i)5-s + (0.656 + 1.47i)6-s + (0.405 − 2.61i)7-s + 2.91·8-s + (−2.00 + 2.22i)9-s + (−1.11 + 1.75i)10-s + (−1.05 − 0.611i)11-s + (0.796 + 1.78i)12-s + (1.31 − 2.28i)13-s + (−0.378 + 2.43i)14-s + (−3.83 − 0.572i)15-s − 0.458·16-s + (−6.76 + 3.90i)17-s + ⋯
L(s)  = 1  − 0.659·2-s + (−0.406 − 0.913i)3-s − 0.565·4-s + (0.537 − 0.843i)5-s + (0.268 + 0.602i)6-s + (0.153 − 0.988i)7-s + 1.03·8-s + (−0.669 + 0.742i)9-s + (−0.354 + 0.555i)10-s + (−0.319 − 0.184i)11-s + (0.229 + 0.516i)12-s + (0.365 − 0.633i)13-s + (−0.101 + 0.651i)14-s + (−0.989 − 0.147i)15-s − 0.114·16-s + (−1.64 + 0.947i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.163i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 + 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.986 + 0.163i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (194, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.986 + 0.163i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0432100 - 0.525269i\)
\(L(\frac12)\) \(\approx\) \(0.0432100 - 0.525269i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.704 + 1.58i)T \)
5 \( 1 + (-1.20 + 1.88i)T \)
7 \( 1 + (-0.405 + 2.61i)T \)
good2 \( 1 + 0.932T + 2T^{2} \)
11 \( 1 + (1.05 + 0.611i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.31 + 2.28i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (6.76 - 3.90i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.02 - 1.16i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.49 + 6.04i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.69 - 3.28i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.13iT - 31T^{2} \)
37 \( 1 + (-0.395 - 0.228i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.67 + 4.63i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.97 + 4.02i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 4.57iT - 47T^{2} \)
53 \( 1 + (-1.18 - 2.04i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 6.82T + 59T^{2} \)
61 \( 1 + 0.292iT - 61T^{2} \)
67 \( 1 + 14.7iT - 67T^{2} \)
71 \( 1 + 2.44iT - 71T^{2} \)
73 \( 1 + (3.51 + 6.09i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 11.5T + 79T^{2} \)
83 \( 1 + (1.79 - 1.03i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.840 + 1.45i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.93 - 5.09i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.79733037225997012764919606526, −10.52877138029768183417365573534, −9.121859715682037645226994303945, −8.362028760678779435607452979823, −7.62985257365584891773592328808, −6.39033255040578267044278579688, −5.26542446249110568182170084689, −4.17858295598874056649623613312, −1.80890990222623096516908756678, −0.50496610188129188550333996716, 2.35416386978439694959293389070, 4.00079098424402811120756819494, 5.17052685322199207862006975739, 6.09075257081495676341807312804, 7.39908978940141068913458537739, 8.761193137039434661121947220802, 9.454630369498577116587062340328, 9.916946398027171157142031694856, 11.29388948984452516911533730168, 11.41167431317567311952579514808

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