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Properties

Label 2-315-105.2-c1-0-14
Degree $2$
Conductor $315$
Sign $-0.538 - 0.842i$
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  + (−0.604 − 2.25i)2-s + (−2.99 + 1.72i)4-s + (−0.254 − 2.22i)5-s + (−2.19 − 1.48i)7-s + (2.40 + 2.40i)8-s + (−4.85 + 1.91i)10-s + (2.61 − 1.51i)11-s + (−1.77 + 1.77i)13-s + (−2.01 + 5.84i)14-s + (0.512 − 0.888i)16-s + (−3.73 − 1.00i)17-s + (3.79 + 2.19i)19-s + (4.59 + 6.20i)20-s + (−4.98 − 4.98i)22-s + (−7.23 + 1.93i)23-s + ⋯
L(s)  = 1  + (−0.427 − 1.59i)2-s + (−1.49 + 0.863i)4-s + (−0.113 − 0.993i)5-s + (−0.828 − 0.559i)7-s + (0.849 + 0.849i)8-s + (−1.53 + 0.606i)10-s + (0.788 − 0.455i)11-s + (−0.492 + 0.492i)13-s + (−0.538 + 1.56i)14-s + (0.128 − 0.222i)16-s + (−0.906 − 0.243i)17-s + (0.871 + 0.503i)19-s + (1.02 + 1.38i)20-s + (−1.06 − 1.06i)22-s + (−1.50 + 0.403i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.538 - 0.842i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.538 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.538 - 0.842i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.538 - 0.842i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.279742 + 0.510834i\)
\(L(\frac12)\) \(\approx\) \(0.279742 + 0.510834i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (0.254 + 2.22i)T \)
7 \( 1 + (2.19 + 1.48i)T \)
good2 \( 1 + (0.604 + 2.25i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (-2.61 + 1.51i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.77 - 1.77i)T - 13iT^{2} \)
17 \( 1 + (3.73 + 1.00i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3.79 - 2.19i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (7.23 - 1.93i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 1.25T + 29T^{2} \)
31 \( 1 + (2.64 + 4.57i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-7.36 + 1.97i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 10.7iT - 41T^{2} \)
43 \( 1 + (-5.73 + 5.73i)T - 43iT^{2} \)
47 \( 1 + (2.09 + 7.80i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.472 - 1.76i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (2.97 + 5.14i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.71 - 6.42i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.50 + 9.36i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 1.07iT - 71T^{2} \)
73 \( 1 + (2.07 + 0.555i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (5.75 + 3.31i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.25 + 1.25i)T + 83iT^{2} \)
89 \( 1 + (-2.44 + 4.22i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.69 - 2.69i)T + 97iT^{2} \)
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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

−11.16881195428139344578651542080, −10.00836903876978127792242529032, −9.421269489079298201450069834036, −8.742583275716744798064677010637, −7.45514806952822545266683007886, −5.96567005852172218268878043169, −4.30444705344075687230461182566, −3.64176657473352878828391251471, −1.99825781338776767612209890369, −0.47562382812395107922049132196, 2.81534125051070973207152652000, 4.47105795975327868620318793930, 5.94627831056808104618805718816, 6.49895741257655291192568246128, 7.32837701764393021379417845020, 8.250280838787742003391053202256, 9.435406171443605424190770572884, 9.884167648907983399945880930100, 11.27412616913283735715607503211, 12.33201447269047186023086086212

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