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Properties

Label 2-315-63.5-c1-0-13
Degree $2$
Conductor $315$
Sign $0.996 - 0.0836i$
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  − 0.396i·2-s + (−1.27 + 1.17i)3-s + 1.84·4-s + (−0.5 − 0.866i)5-s + (0.464 + 0.506i)6-s + (2.43 + 1.03i)7-s − 1.52i·8-s + (0.260 − 2.98i)9-s + (−0.343 + 0.198i)10-s + (−1.94 − 1.12i)11-s + (−2.35 + 2.15i)12-s + (3.23 + 1.86i)13-s + (0.409 − 0.967i)14-s + (1.65 + 0.520i)15-s + 3.07·16-s + (3.44 + 5.96i)17-s + ⋯
L(s)  = 1  − 0.280i·2-s + (−0.737 + 0.675i)3-s + 0.921·4-s + (−0.223 − 0.387i)5-s + (0.189 + 0.206i)6-s + (0.920 + 0.390i)7-s − 0.539i·8-s + (0.0868 − 0.996i)9-s + (−0.108 + 0.0627i)10-s + (−0.585 − 0.338i)11-s + (−0.679 + 0.622i)12-s + (0.896 + 0.517i)13-s + (0.109 − 0.258i)14-s + (0.426 + 0.134i)15-s + 0.769·16-s + (0.835 + 1.44i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0836i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.996 - 0.0836i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.996 - 0.0836i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36988 + 0.0574292i\)
\(L(\frac12)\) \(\approx\) \(1.36988 + 0.0574292i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.27 - 1.17i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-2.43 - 1.03i)T \)
good2 \( 1 + 0.396iT - 2T^{2} \)
11 \( 1 + (1.94 + 1.12i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.23 - 1.86i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.44 - 5.96i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.97 + 1.14i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.69 + 2.70i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.44 + 0.837i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.01iT - 31T^{2} \)
37 \( 1 + (3.18 - 5.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.57 - 2.72i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.90 + 6.76i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 6.66T + 47T^{2} \)
53 \( 1 + (3.20 - 1.84i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 3.23T + 59T^{2} \)
61 \( 1 + 11.2iT - 61T^{2} \)
67 \( 1 + 12.1T + 67T^{2} \)
71 \( 1 + 3.37iT - 71T^{2} \)
73 \( 1 + (10.3 - 5.98i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 5.24T + 79T^{2} \)
83 \( 1 + (8.61 + 14.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.58 - 4.46i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.94 + 2.85i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.52651428894830188062621721710, −10.85645874839007848249471665211, −10.23424595850291629679730589676, −8.821153037279130271897334339747, −8.007240443609342078332211258025, −6.55571621071190974369129287051, −5.71925104551668735647215102044, −4.60408698154267824394326823681, −3.34584093370923012725939615448, −1.50184713191619775394930418643, 1.43546835712098015448003470929, 2.97872852355036324939915602212, 4.94756704926808689880117801357, 5.79240369869872483394877989346, 7.01946305627081228727868891033, 7.50005911128113703739096915360, 8.352050836247364810328949349661, 10.23729588212312317330331372992, 10.94835407188850396609654215629, 11.51245085356941695941716075828

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