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Properties

Label 2-315-315.4-c1-0-18
Degree $2$
Conductor $315$
Sign $0.990 + 0.138i$
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  + (0.866 + 0.5i)2-s − 1.73·3-s + (−0.500 − 0.866i)4-s + (1 + 2i)5-s + (−1.49 − 0.866i)6-s + (−0.866 − 2.5i)7-s − 3i·8-s + 2.99·9-s + (−0.133 + 2.23i)10-s + 6·11-s + (0.866 + 1.49i)12-s + (3.46 + 2i)13-s + (0.500 − 2.59i)14-s + (−1.73 − 3.46i)15-s + (0.500 − 0.866i)16-s + (1.73 + i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s − 1.00·3-s + (−0.250 − 0.433i)4-s + (0.447 + 0.894i)5-s + (−0.612 − 0.353i)6-s + (−0.327 − 0.944i)7-s − 1.06i·8-s + 0.999·9-s + (−0.0423 + 0.705i)10-s + 1.80·11-s + (0.250 + 0.433i)12-s + (0.960 + 0.554i)13-s + (0.133 − 0.694i)14-s + (−0.447 − 0.894i)15-s + (0.125 − 0.216i)16-s + (0.420 + 0.242i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.138i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.990 + 0.138i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.990 + 0.138i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39750 - 0.0972266i\)
\(L(\frac12)\) \(\approx\) \(1.39750 - 0.0972266i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73T \)
5 \( 1 + (-1 - 2i)T \)
7 \( 1 + (0.866 + 2.5i)T \)
good2 \( 1 + (-0.866 - 0.5i)T + (1 + 1.73i)T^{2} \)
11 \( 1 - 6T + 11T^{2} \)
13 \( 1 + (-3.46 - 2i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.73 - i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3 + 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 3iT - 23T^{2} \)
29 \( 1 + (1 + 1.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.92 + 4i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (1 - 1.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.59 - 1.5i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (10.3 + 6i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.06 - 3.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (7 - 12.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.46 + 2i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.5 + 2.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.73 - i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.45167613503392257718259996767, −10.81800718181413965024794411840, −9.948825466356202741555395512045, −9.139035074767230041631409817874, −7.09367289743131409736127585181, −6.48596715555925421609951759651, −6.05919864597650368186255772752, −4.48740609099643483848199833616, −3.78787027442736491461540629094, −1.20152323786165677562876649206, 1.56805648833348602971446120159, 3.59711992163194037596227660269, 4.57651093879000644031405927739, 5.76042371951542005938839939203, 6.22192516838903531164391268284, 8.013103161537253337598518569705, 8.995457197949163602598846838492, 9.751983769598435758543832648214, 11.18152056326321066709302854417, 11.96582916132347260725333370799

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