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Properties

Label 2-315-105.23-c1-0-13
Degree $2$
Conductor $315$
Sign $0.834 + 0.550i$
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  + (2.25 − 0.604i)2-s + (2.99 − 1.72i)4-s + (2.05 − 0.890i)5-s + (−1.48 + 2.19i)7-s + (2.40 − 2.40i)8-s + (4.08 − 3.24i)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 + (1.00 − 3.73i)17-s + (−3.79 − 2.19i)19-s + (4.59 − 6.20i)20-s + (−4.98 + 4.98i)22-s + (1.93 + 7.23i)23-s + ⋯
L(s)  = 1  + (1.59 − 0.427i)2-s + (1.49 − 0.863i)4-s + (0.917 − 0.398i)5-s + (−0.559 + 0.828i)7-s + (0.849 − 0.849i)8-s + (1.29 − 1.02i)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 + (0.243 − 0.906i)17-s + (−0.871 − 0.503i)19-s + (1.02 − 1.38i)20-s + (−1.06 + 1.06i)22-s + (0.403 + 1.50i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 + 0.550i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.834 + 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.834 + 0.550i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.834 + 0.550i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.93211 - 0.879819i\)
\(L(\frac12)\) \(\approx\) \(2.93211 - 0.879819i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.05 + 0.890i)T \)
7 \( 1 + (1.48 - 2.19i)T \)
good2 \( 1 + (-2.25 + 0.604i)T + (1.73 - i)T^{2} \)
11 \( 1 + (2.61 - 1.51i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.77 + 1.77i)T + 13iT^{2} \)
17 \( 1 + (-1.00 + 3.73i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (3.79 + 2.19i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.93 - 7.23i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 1.25T + 29T^{2} \)
31 \( 1 + (2.64 + 4.57i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.97 + 7.36i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 10.7iT - 41T^{2} \)
43 \( 1 + (-5.73 - 5.73i)T + 43iT^{2} \)
47 \( 1 + (-7.80 + 2.09i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.76 - 0.472i)T + (45.8 + 26.5i)T^{2} \)
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 \( 1 + (9.36 + 2.50i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 1.07iT - 71T^{2} \)
73 \( 1 + (-0.555 + 2.07i)T + (-63.2 - 36.5i)T^{2} \)
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

−11.93187076736702219258499905995, −10.91433136913729195502132185542, −9.840977489601077438995432109835, −9.042196771416329303180590154907, −7.41481185224697543504667367482, −6.10151470476803058346614572186, −5.44384263518640696543619733634, −4.65343213184902369665526372394, −3.01113159459531642457902505083, −2.20967347347288633103275184058, 2.42762211469494885255686100062, 3.59439891543466186701565042351, 4.71916069154435401385066027822, 5.84787515101253883751854736591, 6.54802539500000999645666901102, 7.37791744234415693776306484885, 8.857074513532634633622191282117, 10.39187492423421205852234193299, 10.65682394693612978846812207025, 12.29136442977428527932742802932

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