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Properties

Label 2-315-105.2-c1-0-6
Degree $2$
Conductor $315$
Sign $0.291 - 0.956i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.291 - 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.291 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.291 - 0.956i$
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: \((2,\ 315,\ (\ :1/2),\ 0.291 - 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45360 + 1.07606i\)
\(L(\frac12)\) \(\approx\) \(1.45360 + 1.07606i\)
\(L(\frac{3}{2})\) 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

−12.03836602163532632314065714010, −10.69795421590989310597660077859, −9.939089772316390712974045570674, −9.018101367532372089089486669219, −7.81608173980029483683841645305, −6.76446982399615463879315382023, −5.92491600311090430478955024009, −5.40931447584575622953211616262, −3.62670717622471792156271872231, −1.97145781176314231630826290068, 1.54390069330368152772281245983, 2.90546813724675236940085980672, 3.87953916872888301701989356482, 5.45710937229835328874570835796, 6.53633935147759069745798697927, 7.47800409440325251659576917589, 9.044021540096888744079181697701, 10.11190365771778105523836685116, 10.23028774614654627441208655343, 11.58305456106868907856681700912

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