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Properties

Label 2-315-63.20-c1-0-3
Degree $2$
Conductor $315$
Sign $0.851 - 0.524i$
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  + (−2.29 − 1.32i)2-s + (1.70 + 0.281i)3-s + (2.50 + 4.33i)4-s + (0.5 + 0.866i)5-s + (−3.54 − 2.90i)6-s + (−1.30 + 2.30i)7-s − 7.94i·8-s + (2.84 + 0.963i)9-s − 2.64i·10-s + (0.416 + 0.240i)11-s + (3.05 + 8.11i)12-s + (−2.11 + 1.22i)13-s + (6.03 − 3.54i)14-s + (0.610 + 1.62i)15-s + (−5.51 + 9.54i)16-s − 1.67·17-s + ⋯
L(s)  = 1  + (−1.62 − 0.935i)2-s + (0.986 + 0.162i)3-s + (1.25 + 2.16i)4-s + (0.223 + 0.387i)5-s + (−1.44 − 1.18i)6-s + (−0.493 + 0.869i)7-s − 2.81i·8-s + (0.947 + 0.321i)9-s − 0.836i·10-s + (0.125 + 0.0724i)11-s + (0.881 + 2.34i)12-s + (−0.587 + 0.339i)13-s + (1.61 − 0.946i)14-s + (0.157 + 0.418i)15-s + (−1.37 + 2.38i)16-s − 0.406·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.851 - 0.524i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.851 - 0.524i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.760359 + 0.215641i\)
\(L(\frac12)\) \(\approx\) \(0.760359 + 0.215641i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.70 - 0.281i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (1.30 - 2.30i)T \)
good2 \( 1 + (2.29 + 1.32i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (-0.416 - 0.240i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.11 - 1.22i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.67T + 17T^{2} \)
19 \( 1 - 7.56iT - 19T^{2} \)
23 \( 1 + (2.87 - 1.66i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.15 + 1.24i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-8.09 + 4.67i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.34T + 37T^{2} \)
41 \( 1 + (-4.37 - 7.58i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.73 + 9.93i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.433 - 0.751i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 7.15iT - 53T^{2} \)
59 \( 1 + (-4.89 - 8.48i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.60 + 4.38i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.821 + 1.42i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.44iT - 71T^{2} \)
73 \( 1 - 9.10iT - 73T^{2} \)
79 \( 1 + (-5.20 + 9.02i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.09 + 5.35i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 11.7T + 89T^{2} \)
97 \( 1 + (-12.3 - 7.15i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.62034281735417366677002316471, −10.34428158088794086551070831971, −9.773340979608664302794883351997, −9.189871396328919171846837304948, −8.222130132515045948944404996207, −7.52862101797408623442096602610, −6.26377585704789492899903092581, −3.91947823135948694546470953571, −2.73903914413029056376335986726, −1.92792883904674713898926804830, 0.874417288645322136933950904162, 2.56810102481204943527734093079, 4.60904864397340320593930220220, 6.29132621562746302283924758363, 7.10189114220758469150105283557, 7.81864735079236311378758868262, 8.784503070575017749412902043708, 9.437087515108962727321512485074, 10.13448204684628609513856358698, 11.03871494235545489298284281085

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