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Properties

Label 2-315-35.19-c2-0-25
Degree $2$
Conductor $315$
Sign $-0.832 + 0.553i$
Analytic cond. $8.58312$
Root an. cond. $2.92969$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s + (2.5 − 4.33i)5-s + (5.29 − 4.58i)7-s + 8.66i·8-s + (−7.5 + 4.33i)10-s + (−9.26 − 16.0i)11-s + 18.5·13-s + (−11.9 + 2.29i)14-s + (5.5 − 9.52i)16-s + (−1 − 1.73i)17-s + (19.5 + 11.2i)19-s − 4.99·20-s + 32.0i·22-s + (4.5 + 2.59i)23-s + ⋯
L(s)  = 1  + (−0.750 − 0.433i)2-s + (−0.125 − 0.216i)4-s + (0.5 − 0.866i)5-s + (0.755 − 0.654i)7-s + 1.08i·8-s + (−0.750 + 0.433i)10-s + (−0.841 − 1.45i)11-s + 1.42·13-s + (−0.850 + 0.163i)14-s + (0.343 − 0.595i)16-s + (−0.0588 − 0.101i)17-s + (1.02 + 0.592i)19-s − 0.249·20-s + 1.45i·22-s + (0.195 + 0.112i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.553i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.832 + 0.553i$
Analytic conductor: \(8.58312\)
Root analytic conductor: \(2.92969\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1),\ -0.832 + 0.553i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.311154 - 1.02958i\)
\(L(\frac12)\) \(\approx\) \(0.311154 - 1.02958i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-2.5 + 4.33i)T \)
7 \( 1 + (-5.29 + 4.58i)T \)
good2 \( 1 + (1.5 + 0.866i)T + (2 + 3.46i)T^{2} \)
11 \( 1 + (9.26 + 16.0i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 18.5T + 169T^{2} \)
17 \( 1 + (1 + 1.73i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-19.5 - 11.2i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-4.5 - 2.59i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 37.0T + 841T^{2} \)
31 \( 1 + (33 - 19.0i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (27.7 + 16.0i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 32.0iT - 1.68e3T^{2} \)
43 \( 1 + 64.1iT - 1.84e3T^{2} \)
47 \( 1 + (20.5 - 35.5i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-19.5 + 11.2i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (55.5 - 32.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-48 - 27.7i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (55.5 - 32.0i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 74.0T + 5.04e3T^{2} \)
73 \( 1 + (-18.5 - 32.0i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (52 - 90.0i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 32T + 6.88e3T^{2} \)
89 \( 1 + (-55.5 - 32.0i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 74.0T + 9.40e3T^{2} \)
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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

−10.89739091851346537889676756074, −10.26070358531479212038239882613, −9.005483326561102274993796307626, −8.563624162850099793505029912630, −7.62115538124700124060562086032, −5.71926687789396112148155772451, −5.29824932504846138186492001511, −3.65796623002520754543280338634, −1.71152166422481877463011006527, −0.70323791853387566467666198322, 1.84950398403376303289313090935, 3.36629191311748644036697989500, 4.92566920212973261423211574915, 6.15405717485641521253426838152, 7.28299814287552694741156930766, 7.897715650178885085328792313049, 9.047979827673187197016962055117, 9.730401954511427302913488207312, 10.76402918054228083945680758649, 11.60273598749422801806180090410

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