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Properties

Label 2-315-105.2-c1-0-15
Degree $2$
Conductor $315$
Sign $-0.804 - 0.594i$
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  + (−0.364 − 1.36i)2-s + (0.0145 − 0.00842i)4-s + (−2.23 + 0.0163i)5-s + (−2.58 + 0.573i)7-s + (−2.00 − 2.00i)8-s + (0.837 + 3.03i)10-s + (−1.18 + 0.681i)11-s + (0.106 − 0.106i)13-s + (1.72 + 3.30i)14-s + (−1.98 + 3.43i)16-s + (−7.24 − 1.94i)17-s + (−2.03 − 1.17i)19-s + (−0.0324 + 0.0190i)20-s + (1.35 + 1.35i)22-s + (4.94 − 1.32i)23-s + ⋯
L(s)  = 1  + (−0.257 − 0.961i)2-s + (0.00729 − 0.00421i)4-s + (−0.999 + 0.00732i)5-s + (−0.976 + 0.216i)7-s + (−0.710 − 0.710i)8-s + (0.264 + 0.959i)10-s + (−0.355 + 0.205i)11-s + (0.0294 − 0.0294i)13-s + (0.460 + 0.883i)14-s + (−0.495 + 0.858i)16-s + (−1.75 − 0.471i)17-s + (−0.465 − 0.269i)19-s + (−0.00726 + 0.00426i)20-s + (0.289 + 0.289i)22-s + (1.03 − 0.276i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.804 - 0.594i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.804 - 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.804 - 0.594i$
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: \((2,\ 315,\ (\ :1/2),\ -0.804 - 0.594i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.100559 + 0.305421i\)
\(L(\frac12)\) \(\approx\) \(0.100559 + 0.305421i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 - 0.0163i)T \)
7 \( 1 + (2.58 - 0.573i)T \)
good2 \( 1 + (0.364 + 1.36i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (1.18 - 0.681i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.106 + 0.106i)T - 13iT^{2} \)
17 \( 1 + (7.24 + 1.94i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (2.03 + 1.17i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.94 + 1.32i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 4.97T + 29T^{2} \)
31 \( 1 + (3.40 + 5.89i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (9.92 - 2.65i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 7.03iT - 41T^{2} \)
43 \( 1 + (-2.50 + 2.50i)T - 43iT^{2} \)
47 \( 1 + (-0.560 - 2.09i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.62 + 6.06i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.32 - 2.29i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.85 + 10.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.16 + 11.8i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 4.94iT - 71T^{2} \)
73 \( 1 + (5.43 + 1.45i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (10.4 + 6.05i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.06 + 8.06i)T + 83iT^{2} \)
89 \( 1 + (-1.38 + 2.39i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.68 - 3.68i)T + 97iT^{2} \)
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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

−11.11296917759680098044002602897, −10.40548517731676037645228511578, −9.312057687277175976909868705810, −8.597257629231255198297990448868, −7.09103530563830926809525530247, −6.43273359054750906504144178859, −4.69636952167774050141223932343, −3.43206119729807716108727077900, −2.42551801371653663899637362049, −0.23181204626085657830739335129, 2.83032149208263346282249475888, 4.08931237304479237585515114388, 5.54098899420805640363049498595, 6.88168275329601872433308935951, 7.06880480143323275535078109704, 8.532260592609389859362554698025, 8.848684253964631171248593677255, 10.46650110592290134432643755122, 11.20664745665309659645575269076, 12.29474745386682904963963682967

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