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Properties

Label 2-315-105.2-c1-0-1
Degree $2$
Conductor $315$
Sign $0.789 - 0.613i$
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  + (−0.543 − 2.02i)2-s + (−2.08 + 1.20i)4-s + (−1.61 + 1.54i)5-s + (1.07 + 2.41i)7-s + (0.609 + 0.609i)8-s + (4.01 + 2.43i)10-s + (−4.56 + 2.63i)11-s + (−3.08 + 3.08i)13-s + (4.31 − 3.50i)14-s + (−1.50 + 2.60i)16-s + (5.92 + 1.58i)17-s + (0.715 + 0.413i)19-s + (1.51 − 5.17i)20-s + (7.82 + 7.82i)22-s + (1.33 − 0.359i)23-s + ⋯
L(s)  = 1  + (−0.384 − 1.43i)2-s + (−1.04 + 0.602i)4-s + (−0.722 + 0.691i)5-s + (0.407 + 0.913i)7-s + (0.215 + 0.215i)8-s + (1.26 + 0.771i)10-s + (−1.37 + 0.794i)11-s + (−0.855 + 0.855i)13-s + (1.15 − 0.936i)14-s + (−0.376 + 0.651i)16-s + (1.43 + 0.385i)17-s + (0.164 + 0.0948i)19-s + (0.337 − 1.15i)20-s + (1.66 + 1.66i)22-s + (0.279 − 0.0748i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.789 - 0.613i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.789 - 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.789 - 0.613i$
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: \((2,\ 315,\ (\ :1/2),\ 0.789 - 0.613i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.528315 + 0.181275i\)
\(L(\frac12)\) \(\approx\) \(0.528315 + 0.181275i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.61 - 1.54i)T \)
7 \( 1 + (-1.07 - 2.41i)T \)
good2 \( 1 + (0.543 + 2.02i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (4.56 - 2.63i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.08 - 3.08i)T - 13iT^{2} \)
17 \( 1 + (-5.92 - 1.58i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.715 - 0.413i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.33 + 0.359i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 7.45T + 29T^{2} \)
31 \( 1 + (2.97 + 5.15i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.12 + 0.568i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 2.00iT - 41T^{2} \)
43 \( 1 + (1.73 - 1.73i)T - 43iT^{2} \)
47 \( 1 + (-1.11 - 4.17i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.78 - 6.65i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-4.07 - 7.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.12 + 1.94i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.25 - 4.67i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 9.40iT - 71T^{2} \)
73 \( 1 + (-10.4 - 2.80i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (7.72 + 4.46i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.36 + 2.36i)T + 83iT^{2} \)
89 \( 1 + (-4.69 + 8.13i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.63 - 5.63i)T + 97iT^{2} \)
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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

−11.69008686304129940659150667987, −10.92912861893460060651355737737, −10.05108000626380391209442952853, −9.333553990578088463853026629865, −8.056322865336538469323845008582, −7.29608817055208952834753965238, −5.60911884778910876767163940662, −4.27093501290204521599878394672, −2.94804249808838000438936688527, −2.06203022159057065991045146852, 0.44334264377295833310907462448, 3.34769780875126347097863252459, 5.10309085277425152679668956550, 5.39349297853356615406132608961, 7.11460602741501979270377590427, 7.84708361764197707874015632326, 8.131232026403959303081481482155, 9.420363236211734240279961518092, 10.47246328247614792028918407329, 11.50603368754165879704479398349

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