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Properties

Label 2-10-5.2-c12-0-3
Degree $2$
Conductor $10$
Sign $0.998 + 0.0590i$
Analytic cond. $9.13993$
Root an. cond. $3.02323$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−32 − 32i)2-s + (337. − 337. i)3-s + 2.04e3i·4-s + (2.68e3 + 1.53e4i)5-s − 2.15e4·6-s + (−6.68e3 − 6.68e3i)7-s + (6.55e4 − 6.55e4i)8-s + 3.03e5i·9-s + (4.06e5 − 5.78e5i)10-s + 3.24e6·11-s + (6.90e5 + 6.90e5i)12-s + (3.68e6 − 3.68e6i)13-s + 4.27e5i·14-s + (6.09e6 + 4.28e6i)15-s − 4.19e6·16-s + (−1.35e6 − 1.35e6i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (0.462 − 0.462i)3-s + 0.5i·4-s + (0.171 + 0.985i)5-s − 0.462·6-s + (−0.0567 − 0.0567i)7-s + (0.250 − 0.250i)8-s + 0.571i·9-s + (0.406 − 0.578i)10-s + 1.83·11-s + (0.231 + 0.231i)12-s + (0.762 − 0.762i)13-s + 0.0567i·14-s + (0.535 + 0.376i)15-s − 0.250·16-s + (−0.0561 − 0.0561i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0590i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.998 + 0.0590i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $0.998 + 0.0590i$
Analytic conductor: \(9.13993\)
Root analytic conductor: \(3.02323\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :6),\ 0.998 + 0.0590i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(1.69067 - 0.0499329i\)
\(L(\frac12)\) \(\approx\) \(1.69067 - 0.0499329i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (32 + 32i)T \)
5 \( 1 + (-2.68e3 - 1.53e4i)T \)
good3 \( 1 + (-337. + 337. i)T - 5.31e5iT^{2} \)
7 \( 1 + (6.68e3 + 6.68e3i)T + 1.38e10iT^{2} \)
11 \( 1 - 3.24e6T + 3.13e12T^{2} \)
13 \( 1 + (-3.68e6 + 3.68e6i)T - 2.32e13iT^{2} \)
17 \( 1 + (1.35e6 + 1.35e6i)T + 5.82e14iT^{2} \)
19 \( 1 - 4.54e7iT - 2.21e15T^{2} \)
23 \( 1 + (-3.81e7 + 3.81e7i)T - 2.19e16iT^{2} \)
29 \( 1 - 3.01e8iT - 3.53e17T^{2} \)
31 \( 1 - 6.36e8T + 7.87e17T^{2} \)
37 \( 1 + (-2.34e9 - 2.34e9i)T + 6.58e18iT^{2} \)
41 \( 1 + 8.93e9T + 2.25e19T^{2} \)
43 \( 1 + (2.21e9 - 2.21e9i)T - 3.99e19iT^{2} \)
47 \( 1 + (6.89e9 + 6.89e9i)T + 1.16e20iT^{2} \)
53 \( 1 + (-1.73e10 + 1.73e10i)T - 4.91e20iT^{2} \)
59 \( 1 + 3.49e10iT - 1.77e21T^{2} \)
61 \( 1 + 5.44e9T + 2.65e21T^{2} \)
67 \( 1 + (-3.05e10 - 3.05e10i)T + 8.18e21iT^{2} \)
71 \( 1 + 2.45e11T + 1.64e22T^{2} \)
73 \( 1 + (-8.94e10 + 8.94e10i)T - 2.29e22iT^{2} \)
79 \( 1 - 2.19e11iT - 5.90e22T^{2} \)
83 \( 1 + (-2.63e11 + 2.63e11i)T - 1.06e23iT^{2} \)
89 \( 1 + 5.01e11iT - 2.46e23T^{2} \)
97 \( 1 + (9.22e11 + 9.22e11i)T + 6.93e23iT^{2} \)
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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

−18.22043857509153423278342197064, −16.75949741908021793147588798477, −14.70837288875628044082956715696, −13.44826056943094265472635108089, −11.62068524332795176317296734437, −10.16204707063502345217794267871, −8.355371485981063159669530086208, −6.68365528712917515100775863199, −3.36027056835193422189457837241, −1.57035946279989730676396356486, 1.14224739348786163627656264309, 4.19077945153981420794657847820, 6.42487012891694691438688732421, 8.784600294051326754545081659573, 9.424677608240529650374744000215, 11.77338974959768301149881970567, 13.79002084709004101959991506974, 15.19391719578605324413123622868, 16.51332385929637027390145909880, 17.57648209641206247196702886169

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