Properties

Label 2-315-63.38-c1-0-24
Degree $2$
Conductor $315$
Sign $0.295 + 0.955i$
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  − 1.34i·2-s + (1.51 + 0.844i)3-s + 0.186·4-s + (0.5 − 0.866i)5-s + (1.13 − 2.03i)6-s + (−0.829 − 2.51i)7-s − 2.94i·8-s + (1.57 + 2.55i)9-s + (−1.16 − 0.673i)10-s + (−4.07 + 2.35i)11-s + (0.281 + 0.157i)12-s + (4.03 − 2.33i)13-s + (−3.38 + 1.11i)14-s + (1.48 − 0.887i)15-s − 3.59·16-s + (−2.39 + 4.14i)17-s + ⋯
L(s)  = 1  − 0.952i·2-s + (0.873 + 0.487i)3-s + 0.0930·4-s + (0.223 − 0.387i)5-s + (0.464 − 0.831i)6-s + (−0.313 − 0.949i)7-s − 1.04i·8-s + (0.524 + 0.851i)9-s + (−0.368 − 0.212i)10-s + (−1.22 + 0.709i)11-s + (0.0812 + 0.0453i)12-s + (1.12 − 0.646i)13-s + (−0.904 + 0.298i)14-s + (0.384 − 0.229i)15-s − 0.898·16-s + (−0.579 + 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.295 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.295 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.295 + 0.955i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.295 + 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51326 - 1.11629i\)
\(L(\frac12)\) \(\approx\) \(1.51326 - 1.11629i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.51 - 0.844i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (0.829 + 2.51i)T \)
good2 \( 1 + 1.34iT - 2T^{2} \)
11 \( 1 + (4.07 - 2.35i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.03 + 2.33i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.39 - 4.14i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.17 + 1.25i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.68 - 2.70i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.970 + 0.560i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.67iT - 31T^{2} \)
37 \( 1 + (0.507 + 0.878i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.36 + 7.56i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.74 - 4.76i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 12.2T + 47T^{2} \)
53 \( 1 + (-11.6 - 6.74i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 5.15T + 59T^{2} \)
61 \( 1 + 2.54iT - 61T^{2} \)
67 \( 1 - 11.4T + 67T^{2} \)
71 \( 1 - 4.31iT - 71T^{2} \)
73 \( 1 + (3.32 + 1.92i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 6.41T + 79T^{2} \)
83 \( 1 + (-1.81 + 3.14i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.794 + 1.37i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.58 - 2.07i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.08446097644136442025725683094, −10.49771802453691409607091331459, −9.941065548109493955915445305775, −8.860730403302894885377301398381, −7.80143584767311092613555855021, −6.80033746595870857022525221240, −5.09703008875366814843010349183, −3.85276766355810547650823044595, −2.98791785672428636533523970409, −1.55082564186282220996347030635, 2.25743090853329686727872898613, 3.17341617783778568547321819931, 5.21562976597992381792050512285, 6.26372558597165527045952747736, 6.93510249346877490070733299728, 8.075570912626592726717491856232, 8.659116203716227912244416968572, 9.635157982348328619840288760024, 11.06542243899337016870533650889, 11.78962603568319228854337214503

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