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Properties

Label 2-315-63.25-c1-0-17
Degree $2$
Conductor $315$
Sign $-0.107 + 0.994i$
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  + 0.297·2-s + (−1.63 + 0.582i)3-s − 1.91·4-s + (0.5 − 0.866i)5-s + (−0.485 + 0.173i)6-s + (1.24 + 2.33i)7-s − 1.16·8-s + (2.32 − 1.90i)9-s + (0.148 − 0.257i)10-s + (−2.42 − 4.19i)11-s + (3.11 − 1.11i)12-s + (−2.59 − 4.49i)13-s + (0.371 + 0.693i)14-s + (−0.310 + 1.70i)15-s + 3.47·16-s + (1.80 − 3.11i)17-s + ⋯
L(s)  = 1  + 0.210·2-s + (−0.941 + 0.336i)3-s − 0.955·4-s + (0.223 − 0.387i)5-s + (−0.198 + 0.0707i)6-s + (0.471 + 0.881i)7-s − 0.411·8-s + (0.773 − 0.633i)9-s + (0.0470 − 0.0814i)10-s + (−0.730 − 1.26i)11-s + (0.900 − 0.321i)12-s + (−0.719 − 1.24i)13-s + (0.0992 + 0.185i)14-s + (−0.0802 + 0.439i)15-s + 0.869·16-s + (0.436 − 0.756i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.107 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.107 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.107 + 0.994i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.107 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.394189 - 0.439032i\)
\(L(\frac12)\) \(\approx\) \(0.394189 - 0.439032i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.63 - 0.582i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-1.24 - 2.33i)T \)
good2 \( 1 - 0.297T + 2T^{2} \)
11 \( 1 + (2.42 + 4.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.59 + 4.49i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.80 + 3.11i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.03 + 3.51i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.491 - 0.851i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.30 + 3.98i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.20T + 31T^{2} \)
37 \( 1 + (-3.25 - 5.63i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.298 + 0.516i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.0565 + 0.0979i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 7.15T + 47T^{2} \)
53 \( 1 + (-4.78 + 8.29i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 7.50T + 59T^{2} \)
61 \( 1 + 14.9T + 61T^{2} \)
67 \( 1 + 7.57T + 67T^{2} \)
71 \( 1 - 10.2T + 71T^{2} \)
73 \( 1 + (1.67 - 2.90i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 5.18T + 79T^{2} \)
83 \( 1 + (5.45 - 9.45i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.64 + 13.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.85 + 11.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.47754539901164240693569053457, −10.50185961549159217741369665011, −9.569934360153549499720926037326, −8.701131391713209071962016585626, −7.73743796254133995058320837417, −5.90690549782522212334362378971, −5.38420203057218878581794341034, −4.66524358934866043254074503628, −3.03305036598693167300416041723, −0.46282027657589078937350898493, 1.78813116806953901099154265892, 4.10964498366351069178621004575, 4.78295622976018714789984177098, 5.89141351965310647565576098673, 7.14503228321402622718278863258, 7.78275420841482526797342576768, 9.315954521134818779650603961189, 10.28459274403989082520607959457, 10.78065412411894493269656819595, 12.23898830854554882028134178318

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