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Properties

Label 2-315-63.5-c1-0-15
Degree $2$
Conductor $315$
Sign $0.997 + 0.0690i$
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  − 0.917i·2-s + (−0.321 + 1.70i)3-s + 1.15·4-s + (0.5 + 0.866i)5-s + (1.56 + 0.295i)6-s + (0.697 − 2.55i)7-s − 2.89i·8-s + (−2.79 − 1.09i)9-s + (0.794 − 0.458i)10-s + (2.88 + 1.66i)11-s + (−0.372 + 1.96i)12-s + (2.08 + 1.20i)13-s + (−2.34 − 0.639i)14-s + (−1.63 + 0.572i)15-s − 0.345·16-s + (0.514 + 0.890i)17-s + ⋯
L(s)  = 1  − 0.649i·2-s + (−0.185 + 0.982i)3-s + 0.578·4-s + (0.223 + 0.387i)5-s + (0.637 + 0.120i)6-s + (0.263 − 0.964i)7-s − 1.02i·8-s + (−0.930 − 0.365i)9-s + (0.251 − 0.145i)10-s + (0.869 + 0.501i)11-s + (−0.107 + 0.568i)12-s + (0.578 + 0.333i)13-s + (−0.626 − 0.171i)14-s + (−0.422 + 0.147i)15-s − 0.0863·16-s + (0.124 + 0.216i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0690i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0690i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.997 + 0.0690i$
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: \((2,\ 315,\ (\ :1/2),\ 0.997 + 0.0690i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57349 - 0.0544219i\)
\(L(\frac12)\) \(\approx\) \(1.57349 - 0.0544219i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.321 - 1.70i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.697 + 2.55i)T \)
good2 \( 1 + 0.917iT - 2T^{2} \)
11 \( 1 + (-2.88 - 1.66i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.08 - 1.20i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.514 - 0.890i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.75 - 2.74i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.46 - 3.73i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.78 - 2.18i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.525iT - 31T^{2} \)
37 \( 1 + (-3.83 + 6.64i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.0682 + 0.118i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.65 + 9.79i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9.02T + 47T^{2} \)
53 \( 1 + (1.24 - 0.716i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 0.639T + 59T^{2} \)
61 \( 1 - 10.2iT - 61T^{2} \)
67 \( 1 + 4.05T + 67T^{2} \)
71 \( 1 + 4.03iT - 71T^{2} \)
73 \( 1 + (-5.39 + 3.11i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 2.55T + 79T^{2} \)
83 \( 1 + (-2.30 - 3.99i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.33 + 12.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (14.8 - 8.58i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.51484099819606940278161586610, −10.71029954254102162546496724085, −10.00378037079795398299533591277, −9.346120222494711191760477129563, −7.77026098710851939582933958304, −6.73089424088533973358206246091, −5.68088215297344206111342041446, −4.04450009196311766391802838222, −3.49456892923366831231214237654, −1.65813655106231817913338201511, 1.56629122672643489971897668607, 2.90990937057509411539113539543, 5.13016436667892352395287658379, 6.04430317624027584872485572636, 6.57996352731789856525293168434, 7.941976369226017284423809330378, 8.395343846035666005528978739828, 9.561922367081337238932557125051, 11.25830711476888042468386551070, 11.62197268112598206410550412152

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