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Properties

Label 2-605-605.34-c1-0-49
Degree $2$
Conductor $605$
Sign $0.947 - 0.320i$
Analytic cond. $4.83094$
Root an. cond. $2.19794$
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  + (2.18 − 0.314i)2-s + 1.37i·3-s + (2.76 − 0.810i)4-s + (1.33 + 1.79i)5-s + (0.431 + 2.99i)6-s + (2.66 − 4.14i)7-s + (1.76 − 0.805i)8-s + 1.11·9-s + (3.47 + 3.50i)10-s + (−3.26 + 0.576i)11-s + (1.11 + 3.78i)12-s + (−0.872 − 2.97i)13-s + (4.51 − 9.89i)14-s + (−2.46 + 1.82i)15-s + (−1.23 + 0.796i)16-s + (−4.57 + 3.96i)17-s + ⋯
L(s)  = 1  + (1.54 − 0.222i)2-s + 0.791i·3-s + (1.38 − 0.405i)4-s + (0.596 + 0.802i)5-s + (0.175 + 1.22i)6-s + (1.00 − 1.56i)7-s + (0.623 − 0.284i)8-s + 0.373·9-s + (1.10 + 1.10i)10-s + (−0.984 + 0.173i)11-s + (0.320 + 1.09i)12-s + (−0.242 − 0.824i)13-s + (1.20 − 2.64i)14-s + (−0.635 + 0.472i)15-s + (−0.309 + 0.199i)16-s + (−1.10 + 0.961i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.320i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.947 - 0.320i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.947 - 0.320i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.73207 + 0.614433i\)
\(L(\frac12)\) \(\approx\) \(3.73207 + 0.614433i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.33 - 1.79i)T \)
11 \( 1 + (3.26 - 0.576i)T \)
good2 \( 1 + (-2.18 + 0.314i)T + (1.91 - 0.563i)T^{2} \)
3 \( 1 - 1.37iT - 3T^{2} \)
7 \( 1 + (-2.66 + 4.14i)T + (-2.90 - 6.36i)T^{2} \)
13 \( 1 + (0.872 + 2.97i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (4.57 - 3.96i)T + (2.41 - 16.8i)T^{2} \)
19 \( 1 + (-3.01 + 3.47i)T + (-2.70 - 18.8i)T^{2} \)
23 \( 1 + (-1.45 - 2.26i)T + (-9.55 + 20.9i)T^{2} \)
29 \( 1 + (1.70 - 1.97i)T + (-4.12 - 28.7i)T^{2} \)
31 \( 1 + (9.06 + 2.66i)T + (26.0 + 16.7i)T^{2} \)
37 \( 1 + (-1.37 + 4.69i)T + (-31.1 - 20.0i)T^{2} \)
41 \( 1 + (0.0332 + 0.231i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (-2.50 + 1.14i)T + (28.1 - 32.4i)T^{2} \)
47 \( 1 + (-7.64 - 1.09i)T + (45.0 + 13.2i)T^{2} \)
53 \( 1 + (3.36 - 5.24i)T + (-22.0 - 48.2i)T^{2} \)
59 \( 1 + (-0.549 + 3.82i)T + (-56.6 - 16.6i)T^{2} \)
61 \( 1 + (0.549 - 3.82i)T + (-58.5 - 17.1i)T^{2} \)
67 \( 1 + (5.53 - 0.796i)T + (64.2 - 18.8i)T^{2} \)
71 \( 1 + (-7.23 + 8.34i)T + (-10.1 - 70.2i)T^{2} \)
73 \( 1 + (-1.27 - 1.98i)T + (-30.3 + 66.4i)T^{2} \)
79 \( 1 + (-6.63 + 14.5i)T + (-51.7 - 59.7i)T^{2} \)
83 \( 1 + (5.11 - 7.95i)T + (-34.4 - 75.4i)T^{2} \)
89 \( 1 + (6.27 + 7.23i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + (-4.05 + 1.85i)T + (63.5 - 73.3i)T^{2} \)
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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

−10.82837304833607001888869429281, −10.41680889594627742919071017110, −9.241281198128445651276944744658, −7.55936483869158374476148326098, −7.08796643477942989042737942989, −5.70163734737667415854998530707, −4.94491187278865196187625653747, −4.15563232254258338624825316820, −3.32182690614288172891689188672, −1.99470246225188351264458374290, 1.88729914812727050693032971992, 2.56596604112762250204188261433, 4.42107698995655698889524757823, 5.17364619495982150108840162210, 5.70519963604298581342032901653, 6.73017338848835121416697888894, 7.74236282543258567750451158766, 8.758938968566977907217128058651, 9.544507765291660763638476131628, 11.16441642390157103363291230642

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